The United Nations Security Council unanimously voted earlier to refer the brutal repression of the popular Libyan uprising to the International Criminal Court.
Sanctions also included a freeze on the assets of Mr Gaddafi and his immediate family and a global travel ban on the embattled ruler and his close associates.
Mr Hague also said the UK was "working intensively" to establish how many Britons remained in Libya as final evacuation missions were being planned.
Dozens of oil workers were dramatically rescued by military aircraft from remote desert locations last night in a secret mission involving British special forces.
The Foreign Secretary said the mission was launched without the permission of the Gaddafi administration and would not speculate on whether further such operations were planned.
Asked if the Libyan leader could remain in power, Mr Hague told BBC1's Andrew Marr Show: "We have here a country descending into civil war, with atrocious scenes of killing of protesters and a government actually making war on its own so of course it is time for Colonel Gaddafi to go. That is the best hope for Libya.
"And last night I signed a directive revoking his diplomatic immunity in the United Kingdom but also the diplomatic immunity of his sons, his family, his household.
"So it is very clear where we stand on his status as the head of state."
The Foreign Office believes "very few" Britons remain in the capital Tripoli and second city Benghazi - where HMS Cumberland returns today to pick up any more evacuees.
But upwards of 300 are thought still to be in remote desert oil areas - with fresh military-based rescue missions thought to be planned.
"All I can say at the moment is that we are working intensively to establish who is still in Libya and where they are to see how we can assist with getting them out of there. We continue to urge British nationals to leave Libya," Mr Hague said.
The FCO - via its website and social networking and microblogging sites - is informing those seeking to get out of some of the options facing them.
Hungary is sending a plane to Tripoli this morning, potentially with room for some British nationals and the road to Tunisia can be taken with "extreme caution" in daylight, it said.
Mr Hague conceded that the Foreign Office had a "very bad day" on Wednesday when a series of problems saw the first rescue flight stuck on the runway in the UK for many hours.
But he paid tribute to staff based at Tripoli airport who did a "fantastic job".
The Government had "functioned very well" during the crisis despite both Prime Minister David Cameron and his Deputy Nick Clegg being overseas for some of the time, he insisted.
Mr Hague confirmed that ex-prime minister Tony Blair had informed the Government of telephone conversations he held with Gaddafi over recent days.
Reports suggested the Foreign Office had asked him to tell the leader to step down but Mr Hague said: "We are not going to get into a negotiation with Colonel Gaddafi."
And he defended the former premier's initiative to offer the "hand of friendship" to the regime in the 1990s - although not the moves which led to the freedom of the Lockerbie bomber.
"It was right to try to establish a relationship with the Gaddafi government that took Libya away from weapons of mass destruction programmes and the state-sponsorship of international terrorism. If we hadn't done that we might have been in a worse position now." he said.
Britain had to "keep a distance from a dictator", he said.
"We do have to do business though with countries we disagree with. And we still have to do it. I called the Libyan foreign minister last night because you still have to communicate with them personally that this situation is unacceptable."
At least 1,000 people are thought to have been killed in the bloody repression so far.
Mr Hague said the international community had to work closely together to ensure the uprisings across the Middle East were turned into a positive change.
"If we get this right over the coming months it will the greatest advance in world affairs since central and eastern Europe changed so dramatically 20 years ago," he said.
"If we get it wrong well then uncontrolled migration, a breeding ground for international terrorism. These will be the problems coming at us in future years."
The Foreign Office said that planes carrying rescued oil workers were expected to leave Malta at 2pm and 5pm British time, bound for Gatwick, though the exact timings were not confirmed yet.
Gatwick said the first plane was expected at around 5pm.
Today, Deputy Prime Minister Nick Clegg said that Mr Cameron had remained in charge last week. He said his comment that he had "forgotten" he was in charge was a "throwaway remark at the end of a newspaper interview in response to a slightly silly question".
Mr Clegg said he believed media estimates that 380 British nationals could remain in Libya were exaggerated.
He told Dermot Murnaghan's show on Sky News he spent one-and-a-half days skiing with his three children but returned when he realised there was a "huge, huge crisis" developing in the region.
He said: "Of course someone is always in charge and we are always, of course, exactly aware of who it is.
"The Prime Minister was on an official visit and, of course, remains Prime Minister throughout the course of the week.
"We have been very open about the fact, the Prime Minister has sort of apologised for the fact, that the initial reaction, because of the delay in the deployment of a charter plane, was clearly not as quick as we wanted and there were many people in Tripoli airport and elsewhere who were having to face immense uncertainty and huge anxiety."
He added that a mixture of British citizens and other nationalities was rescued in last night's mission, saying that the decision at the United Nations to implement a travel ban and asset freezes was "quite unprecedented".
Mr Clegg said: "We are all relieved it has been done successfully and our information is that the vast, vast majority of people, of British people in Libya, either in the desert or elsewhere, who wanted to leave, have left, either through the operation last night in the desert or getting out themselves or through Tripoli or Benghazi.
"It was meticulously planned and extremely well executed, obviously in an environment of great uncertainty in Libya itself."
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