England, set a target of 191 to win, were faltering at 57 for four before lunch on the fifth and final day.
But Cook (79) and Bell (63 not out) shared a fifth-wicket partnership of 132 that saw England, the world's number one ranked Test side, to the brink of victory before left-handed opener Cook was out with two runs needed when caught in the gully off West Indies captain Darren Sammy.
Bell though struck the winning boundary for a win that gave England a 1-0 lead in this three-match series heading into the second Test at Trent Bridge starting Friday.
England's first innings saw Strauss end his 18-month wait for a Test century and the opening batsman added: "It was lovely to score that hundred and when you do that in a winning cause it makes it all the more special."
West Indies -- who hadn't won a Test match outside the Caribbean against major opposition since defeating South Africa in Port Elizabeth in 2007 and came into this game with a record of just two wins in 30 matches -- made England work for their victory and did well to take the game into the fifth day.
England resumed Monday on 10 for two after Kemar Roach had taken two wickets for seven runs in eight balls to remove Strauss and nightwatchman James Anderson on Sunday.
Both Cook and Jonathan Trott had yet to score.
Roach struck again Monday to remove Trott for 13 with a good length ball that squared him up and took the edge with Sammy, diving to his left, holding a good catch at second slip.
And 13 also proved an unlucky number for Trott's fellow South Africa-born batsman Kevin Pietersen.
He had just pulled Test debutant Shannon Gabriel's third delivery of the innings for four when, to the fast bowler's next ball ball, he tried to repeat the stroke and got a bottom edge to wicket-keeper Denesh Ramdin.
England were in trouble but Cook got them going again.
He completed a 78-ball fifty featuring eight fours when he late cut off-spinner Marlon Samuels and after lunch Bell followed him to the landmark in 84 balls.
That West Indies started the last day with even an outside chance of victory owed much to Shivnarine Chanderpaul.
Officially the world's best batsman, the Guyana left-hander made scores of 87 not out and 91 while spending more than 10 hours at the crease in this match.
Together with Samuels (86) he put on 157 for the fifth wicket in the second innings as the West Indies gave their bowlers a target to defend and a chance of a first win in 15 Tests in England.
For England, this Test was a personal triumph for Stuart Broad, the man-of-the-match.
The fast-medium bowler took a Test-best seven for 72 in the first innings and four for 43 in the second for a match haul of 11 for 165.
Broad, who made 169 against Pakistan at Lord's in 2010, became just the fourth player to take five wickets in an innings, 10 in a match and score a century in Lord's Tests.
Only England's Gubby Allen and Ian Botham, and Australia's Keith Miller, had previously completed that 'treble'.
