As the two men met in the Afghan capital, Kabul, the Prime Minster stressed that a leaked US diplomatic cable which quoted Mr Karzai as saying Britain was "not up to the task" in Helmand province dated back to before the surge in troops which has seen the UK deployment increased to 10,000 and joined by 20,000 Americans.
Mr Karzai voiced his "respect" for the British military's bravery and skills during talks today, said Mr Cameron.
The Prime Minister restated his intention to start the transition to Afghan control of parts of the country in 2011 and complete the withdrawal of combat troops by the end of 2014. And he announced agreement on a long-term partnership plan which will see Britain offer economic, political and military support for Afghanistan after that date.
During his surprise two-day pre-Christmas visit to troops, Mr Cameron has voiced "cautious optimism" about the progress of military operations and said British troops could start to be withdrawn next year.
His view on the timetable was backed by the head of the UK Armed Forces, Chief of the Defence Staff General Sir David Richards, who accompanied him on a tour of bases.
Gen Richards hailed the "astronomical" quickening of results on the ground and said success in the mission was "eminently do-able".
Describing himself as a "good friend" of the Prime Minister, Mr Karzai said at a joint press conference in his Kabul presidential palace that there was "some truth and some not-so-truths" in last week's WikiLeaks disclosures.
Asked if he would apologise for his quoted words, he replied: "Britain has been a steadfast supporter of Afghanistan and of the Afghan people. Britain has contributed in the sacrifice of its soldiers, of blood and of resources in Afghanistan, for which the Afghan people are extremely grateful.
"You have been operating as the British Army and trainers and civilians in a very difficult part of the country. We fully understand and appreciate the hard work you have been doing.
"So my word today to the British people on behalf of the Afghan people is gratitude for the sacrifice and the resources that Britain has brought to Afghanistan and the dedication to a stable, strong Afghanistan, a peaceful Afghanistan and an Afghanistan that will be standing on its own feet, rather than being dependent on others."
And Mr Cameron said: "I don't want the WikiLeaks to come between a strong relationship."
He acknowledged that between 2006 and 2008 - when Mr Karzai's reported comments were made - "there weren't enough troops in Helmand province".
With the surge which began last year there was now "a density of forces that are able to deliver security to the vast majority of people", he said.
Speaking alongside Mr Karzai in Kabul, Mr Cameron said he had three priorities for 2011, which he said must be made "a decisive year in this campaign" - to maintain the security momentum created by the military surge; to begin the process of transition of security responsibility to Afghan forces; and to accelerate the Afghan-led political process of integration and reconciliation of insurgents.
Mr Cameron said: "2010 was without doubt a year in which we made real progress.
"2011 must be the year in which that progress becomes irreversible, because a safer Afghanistan means a safer Britain and a safer world."
The Prime Minister said he had seen that the people in Helmand were becoming more confident in returning to their ordinary lives as the surge of troops delivers more security.
He added: "Of course, there is no scope for complacency. This progress is still fragile.
"But I am cautiously optimistic. We have the right strategy... we have put in the right resources to back it up and we have also given it a very clear focus on national security and we are on the right track.
"What I have seen on this visit gives me confidence that our plans for transition are achievable."
Transition to Afghan control would begin early next year and be completed by the end of 2014, as agreed at last month's Nato Lisbon summit, said Mr Cameron.
But he added: "This doesn't mean that the international community will in any way abandon Afghanistan after 2014.
"On the contrary, we made it clear in Lisbon that we will stand by you for the long term. Britain will remain a close and reliable partner and friend for many years to come.
"So I'm pleased to announce that we will work together on a new long-term partnership between Britain and Afghanistan which will set out in black and white the ways in which we will support you politically, economically and militarily... once our combat troops have gone home."
Speaking ahead of his meeting with Mr Karzai, the Prime Minister insisted that the concern exposed by the WikiLeaks website was out of date and "does not change any of the fundamentals".
The cables detail conversations in which Mr Karzai said UK forces were "not up to the task" of securing southern Helmand province and that the job would be better given to the US.
Helmand Governor Gulab Mangal was also said to have criticised UK troops for failing to get out of their bases and engage with local people.
Addressing reporters at Camp Bastion, where he stayed overnight after visiting troops serving in Afghanistan yesterday, Mr Cameron played down the impact of the leaks.
"Of course WikiLeaks has led to lots of embarrassing questions and all the rest of it, but I think, in the end, it does not change any of the fundamentals between Britain and America, it doesn't change any of the fundamentals between Britain, America and Afghanistan, but obviously it has provided a lot of copy," he said.
"It was related to a previous period when we all know now there were not enough troops in Helmand."
Asked how it would affect relations with President Karzai, he said: "Of course sometimes there are frustrations but we want someone who is going to speak up for Afghanistan and who wants to take control of Afghanistan, who wants an Afghanistan without foreign forces on its soil.
"We have frank exchanges, of course we do, but I don't think either of us want this to get in the way of what needs to be a strong relationship between our countries and between our governments in order to deliver what we want."
The Prime Minister was also due to hold talks with America's top military leader in Afghanistan, General David Petraeus, in Kabul before returning to UK.
Gen Richards said: "At no stage were they criticising the ability of the British soldier. It was all about our inability to produce the force ratios - those days are past us and our linkages at every level with the American armed forces are incredibly strong."
Gen Richards had previously discounted the prospects of beginning a drawdown next year but said it was now possible if progress continued.
The two men were speaking after visiting bases and seeing the training of Afghan police and army forces - as part of the traditional pre-Christmas visit by a premier.
Talking to reporters at the UK's main Camp Bastion base in Helmand province, Mr Cameron was asked if he still believed the 2011 date to begin pulling out was feasible.
"The British public want to know that there is an endpoint to this, that we are making progress, that this can be done, and I was keener to do that rather than make absolutely definite noises either about 2010 or 2011 but I do think it's possible," Mr Cameron said.
Gen Richards said: "It is conditions-based next year but looking at the progress we have made - I was only here three months ago - it is quite astronomical how quickly things are coming together."
It was also announced that the UK will invest £135 million in doubling the capability of unmanned drones - used to seek out and destroy roadside bombs and the insurgents behind them.
The cash is expected to pay for around five new remotely-piloted aircraft which will be in service by 2013.
Mr Cameron was also shown Warthog armoured vehicles - the full complement of more than 100 of which are expected to be deployed in Afghanistan within the next few weeks.
The Prime Minister said there would have to be an inquiry to look into a "mistake" by US forces which appeared to have caused the death of a British soldier.
He said the apparent friendly-fire incident, involving a US aircraft, was "very tragic" but a consequence of "the fog of war".
"It is a very tragic case. It is particularly tragic when you have one of these incidents of so-called friendly fire," he told reporters.
"There needs to be an inquiry, as there always is in a case like this, so we get to the bottom of what has happened and why a mistake was made in this case.
"One's heart goes out to the family. It is painful and difficult enough to lose a loved one, without it happening in this way.
"We have got to do everything we can to try and stop this happening in the future but in the fog of war tragically these things do sometimes happen."
Mr Cameron went on: "We should always bear in mind that, whether it is the RAF or whether it is the US Air Force, they do an enormous amount to save lives of our troops in Afghanistan and in combat, but obviously you can't stop trying to learn the lessons of how to stop these sort of things happening in the future."
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