Fast bowler Daren Powell had brought West Indies back into the game with two wickets soon after lunch.
But together with Kevin Pietersen, 26 not out, Cook shared an unbroken stand of 55 in 87 balls to counter Powell who at tea had figures of two wickets for 49 runs in 17 overs.
Left-hander Cook, omitted from England's World Cup squad, was now in sight of a fifth century in his 15th Test before the players were presented to Britain's Queen Elizabeth II and her husband, the Duke of Edinburgh.
England, 85 without loss at the end of the morning session, saw stand-in skipper Andrew Strauss depart in the third over after lunch when he failed to get over the top of a square drive against Powell and was caught by Devon Smith at point for 33, having just added two runs to his interval total.
England were now 88 for one in the 25th over.
New batsman Owais Shah, a Middlesex team-mate of Strauss's and playing his first Test in England, made a nervy start.
But at the other end Cook, 47 not out at lunch, completed a 76-ball fifty including five fours.
Shah's mood would not have been helped after he was hit on the back of helmet by a wayward throw from fast bowler Jerome Taylor while still on nought.
But Shah got off the mark before off-driving Corey Collymore for four to bring up England's hundred.
However, on six, he was squared up by a fine delivery that veered up the Lord's slope and was caught by Smith, now at third slip. Powell had struck again and England, not for the first time, had lost two wickets in quick succession.
Pietersen counter-attacked with a cover-driven four off Powell.
England, sent in by new West Indies captain Ramnaresh Sarwan on a pitch freshened by several days of rain, made a cautious start under overcast skies against Powell and Taylor.
Despite the helpful conditions the new ball duo didn't trouble the batsmen too often, the duo not helped by the fact only West Indies batsmen took the field in their lone warm-up match, against Somerset, where two of the three days were washed out.
Cook, who had been struggling to time the ball, cashed in when he cut a short ball from Collymore to the boundary and then pulled Powell to the rope.
Strauss was leading England in the absence of injured first-choice skipper Michael Vaughan. The Yorkshire batsman, who hasn't played a Test in 18 months mainly because of knee and hamstring problems, was ruled out of this game with a broken finger.
Also missing was Andrew Flintoff, the all-rounder suffering a recurrence of a left ankle problem.
This was England's first Test since their 5-0 Ashes defeat in Australia and first under new coach Peter Moores.
West Indies, who haven't won an overseas Test since 2000, also had a new coach in David Moore.
This was their first major match since the retirement of former captain Brian Lara, Test cricket's leading run scorer, who quit after the World Cup in the Caribbean where both the hosts and England failed to reach the semi-finals.
Brief Score (at tea, D-1):
England 158 for 2 (Cook 82 n o, Strauss 33, Pietersen 26 n o; Powell 2 for 49) Vs West Indies.