Burnley started brightly and created the best chances in the opening 25 minutes, Mickey Mellon firing into the side-netting and Andy Payton almost fooling Mart Poom in the Rams' goal with a delicate chip.
County came into the game later in the first half but Burnley always looked dangerous and it was no surprise when Cooke eventually found the target, leaping to head Gordon Armstrong's deep cross from the left down past Poom. Derby, who had lost seven of their nine Premiership matches at home, were on their way out.
Jim Smith, the Rams' manager, missed the final 15 minutes. He had to catch a flight to a European destination to watch a player he hopes to sign but his assistant Billy McEwan said: "All the players know in no uncertain terms that we are not pleased with this result."
Stan Ternent, Burnley's manager, summed up the match by saying: "It was a thoroughly deserved victory. We played really well and I've always said if we perform to our full capability and potential we are capable of beating anyone. Derby have got good players but are going through a sticky patch. Confidence is a big thing in football and when we got the goal their heads went down and the crowd got on their backs.
"We will enjoy this. It's a fantastic boost for the chairman and the club to come here and beat a Premiership team in a great stadium. It is what the FA Cup is all about."
Elsewhere, Second Division Luton Town fought back to earn a replay at Kenilworth Road against ambitious Fulham, who trailed 1-0 and led 2-1 but were pegged back by Matthew Spring's late equaliser.
Another London First Division side, Queen's Park Rangers, were also pegged back eight minutes from time by lowly opposition, in this case Third Division Torquay United. Stuart Wardley kept up his scoring sequence with an early strike for Rangers but Michael O'Brien's goal takes Rangers back to Devon.
On a good day for the West Country survivors, Second Division Reading must have thought they had seen off Plymouth Argyle's challenge at the Majedski Stadium where they led through Jim McIntyre's goal. But another 82nd-minute equaliser, courtesy of Chris Hargreaves, earned a replay at Home Park. First Division Walsall needed a 75th-minute penalty, converted by Mark Robins, to earn a replay with Second Division Gillingham.
The Norwich City goalkeeper, Andy Marshall, had a nightmare against Coventry City at Carrow Road. With the score goalless he let Noel Whelan's harmless cross slip through his hands for Coventry's first and, after Chris Llewellyn equalised, allowed Cedric Roussel's shot to wriggle under his body. Coventry went on to win 3-1. Portsmouth, who sacked their manager, Alan Ball, last week, lost 1-0 at Sunderland's Stadium of Light.
Bradford City took a day off from their difficulties in adjusting to the quality of play in the Premiership to beat First Division Crewe 2- 1. It was 1-1 when the substitute Dean Saunders popped up to send the Bantams into the next round. Wimbledon beat Barnsley, managed by Dave Bassett, the Dons' former manager, 1-0 at Selhurst Park.
In the games between First Division teams, results went according to league form. Charlton beat struggling Swindon Town 2-1 at the Valley and Blackburn's improved performances under caretaker manager Tony Parkes - unbeaten in seven games - continued when they drew 2-2 at West Brom. Parkes said he could take on the job permanently. Maybe. "I'm a bit more undecided about it. It's always been no, I don't want it, but we'll see what happens in the next 10 days or so."
On the East Coast, Grimsby Town scored two goals in the final two minutes through Steve Livingstone, his second of the game, and Bradley Allen to snatch a 3-2 win over Stockport. At Deepdale, Preston, 22 points above Oldham in the Second Division, won 2-1.Reuse content