At stumps England were 309 for three after Cook, dropped three times, and Collingwood, missed once, both scored their second Test centuries and first at home after they had each reached three figures against India in Nagpur in March.
Left-hander Cook finished on 101 not out and Collingwood 109 not out, their stand worth 221 in 71 overs after England had been 88 for three against a second-string attack.
"It was a great day and great to have a big partnership," said Collingwood.
"We batted a long time in Nagpur and it was nice to do it again today. We had a few tricky moments but we dug in deep and knocked the ball about well, especially in the afternoon session."
Cook admitted he had enjoyed some good fortune.
"Once we got through the new ball, the pitch became quite good to bat on," he said.
"I had some luck and made the most of it."
Collingwood said both he and Cook were delighted to have got on the Lord's honours board which records every Test century scored at the 'home of cricket'.
"We spoke about getting on the board before the game and it's nice to have done that."
Meanwhile, Pakistan bowling coach Waqar Younis admitted the team's bowlers had been denied the support they deserved.
"It was not the best of days for Pakistan," said the legendary fast bowler. "The ball did swing a lot but we didn't really cash in."
At tea, Cook had been 68 not out and Collingwood 77 not out.
Pakistan, who had already missed two chances to catch Cook, then gave Collingwood a reprieve after he had added just two runs to his score when wicket-keeper Kamran Akmal dropped a straightforward catch off Umar Gul.
Akmal, who had injured his left index finger during Pakistan's final warm-up match against England A at Canterbury, immediately threw off his glove and the only recognized keeper in Pakistan's squad seemed in pain.
Then Imran Farhat, who had dropped Cook on nought, again missed him in the slips when the Essex batsman, on 81, edged a drive off fast bowler Mohammad Sami.
Collingwood then went to 99 when he lofted leg-spinner Danish Kaneria high over long-off, a shot that also saw England to a new record fourth-wicket partnership against Pakistan, breaking the 188 of Ted Dexter and Peter Parfitt at Karachi in 1961/62.
And next ball a single off Kaneria took Collingwood to a hundred in 157 balls with 12 fours.
Cook spent a nervous eight balls on 99 before a quick single off Shahid Afridi saw him through to a hundred in 259 balls with nine fours.
But it could easily have been a different story.
Cook, 21, had yet to score when he edged paceman Umar Gul, only for Farhat, diving across from second slip, to drop a catch heading straight to first slip and Pakistan captain Inzamam-ul-Haq.
At that stage England, whose stand-in captain Andrew Strauss won the toss and batted, had slumped to 60 for two.
After lunch Pakistan saw Cook, on 45, check a drive against Kaneria only for the batsman's Essex team-mate to drop a straightforward caught and bowled chance with England 196 for three.
Collingwood went to a 65-ball fifty with his seventh four - a punched backfoot drive through the offside against Sami.
Pakistan had performed creditably in the morning given they were without several injured quicks in Shoaib Akhtar (ankle), Rana Naved-ul-Hasan (groin) and Mohammad Asif (elbow).
And they also had to omit unfit batsmen Shoaib Malik (elbow) and vice-captain Younis Khan (knee).
After openers Marcus Trescothick and Strauss put on 50 at nearly a run-a-ball, England lost both left-handers with the total on 60.
Trescothick (16) went first, edging Gul to Akmal. Then stand-in skipper Strauss, in his first Test as England captain, was lbw for 30 to Abdul Razzaq.
Kevin Pietersen looked in good touch before on 21 he was lbw to Razzaq to leave England wobbling.
Pakistan, who beat England 2-0 at home in a three-match Test series late last year, have not lost a Test series in England since 1982.
England 1st Innings
Fall of wickets: 1-60, 2-60, 3-88
To bat: I Bell, G Jones, L Plunkett, M Hoggard, S Harmison, M Panesar
Pakistan: Salman Butt, Imran Farhat, Faisal Iqbal, Inzamam-ul-Haq (capt), Mohammad Yousuf, Abdul Razzaq, Shahid Afridi, Kamran Akmal (wkt), Mohammad Sami, Umar Gul, Danish Kaneria
Umpires: Steve Bucknor (WIS), Simon Taufel (AUS)
TV umpire: Peter Hartley (ENG)
Match referee: Ranjan Madugalle (SRI)