At lunch here Thursday, England were 85 without loss in the 22 overs possible after rain delayed the start of this four-Test series by 30 minutes.
Strauss was 31 not out off 65 balls with six fours and Cook 47 not out off 67 balls with four boundaries.
England, sent in by new West Indies captain Ramnaresh Sarwan on a pitch freshened by several days of rain, made a cautious start in overcast conditions against new ball duo Daren Powell and Jerome Taylor.
Their openers had a nervy moment when, with the score on 11, Cook set off for a quick single. Strauss, who hesitated, nevertheless beat onrushing point fielder Shivnarine Chanderpaul's underarm direct hit throw.
And in the last over before lunch Strauss, on 31, edged Corey Collymore just short of the slip cordon.
In the sixth over Strauss pulled and leg-glanced Taylor for two fours in as many balls, England's first boundaries off the bat.
Despite the helpful conditions Powell and Taylor 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.
First change Collymore saw his second ball square cut for four by Strauss.
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 too had a new coach in David Moore and 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 lunch, D-1):
England 85/0 (Alastair Cook 47 n o, Andrew Strauss 31 n o) Vs West Indies.