The threat of Kemar Roach (three for 60) continued to sow doubts about the superiority of the world number one team on the final morning at Lord’s.
But, after England had stumbled to 57 for four – in pursuit of 191 to go 1-0 up with two to play – Cook (79) in particular did not panic.
Back-to-form Bell (63 not out) completed his second half-century of the match too, to help seal the outcome by mid-afternoon after Cook cut Darren Sammy straight to gully to go with only two more runs needed.
Roach landed an early blow with the wicket of Jonathan Trott and Shannon Gabriel also soon eliminated Kevin Pietersen.
The clouds, which have persisted almost throughout this seesaw match, were again present as Cook and Trott set out to calm home nerves against Roach and Fidel Edwards with a ball just four overs old.
But Trott got no further than the sixth over of the day and only his second delivery from Roach.
He edged the first low through the slips for four, then Roach got the second to move off the pitch up the slope again, and resulted in a neat catch by Sammy away to his left at second slip.
West Indies, therefore, got their wish – an early look at the dangerous Pietersen with the ball still new. Pietersen responded with a busy approach, and in under four overs he and Cook had gone past 50 from 29 for three.
But the introduction of debutant fast bowler Gabriel brought a last transient twist in the tale. Pietersen had clubbed a pull for four to midwicket from the third ball of Gabriel’s first over. Once again, though, a boundary was instantly followed by a wicket as Pietersen aimed for a repeat to a ball which was slightly less short and took an under-edge for caught behind.
Cook, joined by Bell, gradually found some fluency after a stodgy start on a still benign surface – and England appeared intent on keeping the scoreboard moving, even though time was not going to be an issue.
This was a world away, of course, from their unsightly collapse to 72 all out in the Abu Dhabi desert against Abdur Rehman et al in their low point of the year to date – a 72-run defeat by Pakistan in January.
But if any lesson learned there could be transferred to the home of cricket, it was the folly of allowing caution to preclude even a trickle of the runs required.
As Sammy juggled his options, to give Roach necessary rest, Cook regularly found the boundary with cuts and drives as England moved to within 60 runs of their target by lunch – thereby puncturing the tourists’ belief that they could somehow sneak an unexpected win.
Cook completed a 78-ball 50 in the last over of the session, with a back-cut off Marlon Samuels for his eighth four.
Sammy’s decision to persevere with Samuels’ innocuous off-spin on the resumption was baffling, and did nothing to discomfort Cook or Bell – or provide much support for the returning Roach at the pavilion end.
The route was therefore cleared for an England victory, widely considered a formality before the start of this match but achieved in unpredictable fashion. Only Stuart Broad’s 11-wicket haul swung the verdict, as West Indies fought back with great determination on the third and fourth days.