[Scorecard: Second ODI, Oval]
Cook scored 112 in putting on a 122-run opening stand with Ian Bell (53). He then added 81 with Jonathan Trott, who made 43 not out, to steer the hosts to 239/2 and eclipse West Indies' 238/9 with five overs to spare at The Oval. Cook smashed 13 boundaries and one six in 120 balls.
"It was a really good team performance in some tricky circumstances," Cook said. "Credit to the group for the way we've handled it, it was a really good performance and a comfortable win in the end. "We held our nerve."
"We didn't score enough runs. At the end of the day you need runs on the board to defend. As long as we put on a partnership we kept losing wickets, which set us back. We wanted 280, we know the Oval wicket is a good one. It's always hard to ask the lower order to try and get runs. We didn't bowl our lines and lengths consistently enough but they played well on a good wicket," said West Indies' skipper Darren Sammy.
Sammy was the pick of the West Indies bowlers, with 2-46. Earlier, Chris Gayle lived up to his reputation in his first match for West Indies since the World Cup quarter-finals 15 months ago, and the end of a public spat with national officials.
He slogged England's attack with five sixes and three boundaries in a 51-ball 53 but his harsh demise sparked a four-wicket collapse for 16 runs. Dwayne Bravo (77) and Kieron Pollard (41) led the team back to respectability in a stand of 100, but their exits saw the innings peter out tamely. James Anderson took 2-38 and Stuart Broad 2-43 in an England side unchanged from the one which won the first one-dayer by 114 runs on Saturday.
England won the toss and put West Indies in, although West Indies said it wanted to bat first anyway. The move started well for England as Gayle and Lendl Simmons struggled through the first five overs, putting only 8 on the board.