The contest was tougher than anyone had predicted, but after surviving one final scare from the tourists in the morning, when England were reduced to 57 for four in pursuit of a fourth-innings target of 191, the world’s number one-ranked Test team came through to win.
Kemar Roach was expensive but menacing, accounting for Jonathan Trott with a ball that moved away up the Lord’s slope and drew an outside edge which Darren Sammy caught at slip.
Debutant Shannon Gabriel struck in his first over, with Kevin Pietersen aiming to stamp his authority on proceedings, but only managing to bottom-edge a pull shot to the wicketkeeper.
But Cook (79) and Bell (63 not out) blunted West Indies’ momentum with sensible strokeplay on a pitch which still played well on its fifth day.
And Roach aside, the West Indies bowling attack lacked the menace to prise out the remaining wickets.
England impressed in the way they set about the target, attacking the total with purpose, with the memories of an abortive chase of 145 against Pakistan in the United Arab Emirates, when they folded for 72 all out, still fresh enough in the memory.
Having overcome the shaky start, England added 121 runs at a healthy lick in the morning session to leave just 60 more required in the afternoon.
They added those in double-quick time, with West Indies’ belief finally shaken.
Sammy's team have already acquitted themselves better than they managed in the corresponding series three seasons ago but in a rare misjudgement from the tourists, they persisted with the off spin of Marlon Samuels in the session after lunch, rather than teaming Roach with Fidel Edwards for one final onslaught.
Roach ran in once more with enthusiasm but was blighted by a no-ball issue which had affected him all match long. By the end of the contest, the paceman had a remarkable 18 no balls to his name over the two innings.
Cook fell just two runs short of the target trying to cut a bit too close to his body and handing Sammy a wicket. It ended a partnership worth 132 runs, but made no difference to the final outcome.
Bell drove the winning boundary through midwicket four balls later to give the home side a 1-0 lead going into Friday’s second Test at Trent Bridge.
But having taken that lead, they will know that if West Indies apply themselves as well as they did at the home of cricket, the series is far from over.