Anderson (4-45) had a hand in all six of the West Indies wickets to fall before lunch as the home side slipped from their overnight 202-2 to 282-8, before being bowled out for 307 in their second innings leaving England chasing 143 to win.
Jonathan Trott chopped Shannon Gabriel on for a duck but Cook (59no) and Gary Ballance (81no) shared an unbroken stand of 142 to steer England to their target in comfort off 41.1 overs, closing on 144-1
Both batsmen completed their second half-centuries of the match but the day belonged to Anderson who tilted the Test in England???s favour by claiming three wickets, two catches and a superb run out before lunch.
England failed to add to their wicket tally in the first five overs of the day with the old ball but their fortunes were transformed with the very first delivery with the new one, Kraigg Brathwaite (116) fending Anderson off to Joe Root in the gully.
Anderson had another 14 runs later as Cook produced a brilliant slip catch to send back the out-of-nick Shivnarine Chanderpaul after Ian Bell shelled a straightforward edge at second slip. Cook swooped at second slip as the chance went down, grabbing the ball with his right hand and holding it at the second attempt to make it 238-4.
Just one run had been added to the score when Marlon Samuels was too slow to withdraw his bat and edged Anderson through to wicketkeeper Jos Buttler. Ben Stokes ??? saluted by Samuels when dismissed on day three ??? applauded enthusiastically as he was passed by the departing batsman.
There was to be no keeping Anderson out of the action as he then safely pouched a poorly-executed drive from Jermaine Blackwood at mid-off, Chris Jordan the bowler to benefit from the 23-year-old???s misjudgement.
The Lancashire seamer also had a hand to play in the fifth wicket of the morning to fall, running out non-striker Jason Holder with a direct hit after Gary Ballance had parried a tough chance offered by Ramdin.
Next to go was Kemar Roach, attempting to hit Moeen Ali over mid-on but only succeeded in picking out that man Anderson, who caught the chance running to his right.
Moeen then accounted for Ramdin and Gabriel in the same over soon after lunch to wrap up the innings.
It left England with a target of 143 to chase in 59 overs but they got off to a poor start with Trott falling in the second over ??? the fifth England batsman to be dismissed chopping onto their stumps in the match.
Cook and Ballance steadied things though with an unflinching partnership while West Indies woes deepened as Jason Holder rolled his ankle in his follow through and was stretchered off the field with a lateral ligament strain.
Ballance gave one chance to slip before he reached his fifty off 86 balls with six fours and a six and another with 17 more runs needed, while Cook took 11 more balls to reach the landmark.
Victory was secured when Ballance cut Samuels behind point for four, prompting Cook to punch the air in triumph.