Cook (112) shared a century opening stand with Ian Bell (53) and then put on 81 for the second wicket with Jonathan Trott as England raced past West Indies' patchy 238 for nine with five overs to spare to establish an unassailable 2-0 lead.
The hard work was done, though, when Cook's bowling attack withstood an early barrage from Chris Gayle (53) and restricted the Windies to an under-par total despite a worthy 77 from Dwayne Bravo.
On a typically reliable batting surface of even pace and bounce, Cook and Bell then duly put England's achievable target into context.
There was barely a false shot from either as Cook won the race to 50, until Bell forewent the chance of his own second successive ton by pushing a change of pace from Darren Sammy straight into the hands of cover.
Cook made no mistake, until he lobbed a cutter from Sammy over the bowler to be caught at mid-on, as he extended an enviable sequence of six centuries from an England opener in consecutive one-day internationals – a series he began himself with back-to-back hundreds against Pakistan in Abu Dhabi.
Unsurprisingly, six wins have resulted, Cook underpinning the latest with his fifth hundred at this level – containing 12 fours in 114 balls.
Bravo identified the dismissal of Gayle, playing in his first international match since last year's World Cup, as an obvious turning point.
He made it clear, too, that the West Indies were far from pleased with the outcome of Gayle's review, in which third umpire Kumar Dharmasena appeared to conclude that the simultaneous impact of the ball with both pad and bat vindicated the original decision.
"What confused us is they used the technology . . . and then the decision was given," he said.
"It's okay, umpires do make mistakes – that's accepted – but not when they see it after a referral and realise they've made the wrong decision and stand by it.
"It was a game-changing moment, and it was always difficult then to get a big total."
Once Gayle was gone, the momentum swung emphatically towards England.
"Obviously, the decision did not help us," added Bravo. "Chris got off to a very good start; then unfortunately the way he got out put us on the back foot.
"If the decision had gone a different way, it might have been a different ball game. We lost our way in the middle there. Pollard and I fought our way back, but we were not in a position to have the firepower to finish off the innings."