During the course of his innings Gayle, who has previously indicated this will be his last World Cup, overtook Viv Richards as the leading scorer in one-day internationals between England and the West Indies.
Andre Russell shone briefly but was caught by Woakes off the bowling of Wood for 21 and the last four wickets tumbled for just 10 runs. England are now second in the World Cup table with three wins and one defeat, one point behind New Zealand, while the West Indies have won one, lost two and had one no result in their four matches. Wood finished with figures of 3-18, while Archer took 3-30.
Jofra Archer and Mark Wood took three wickets apiece as England bowled the Windies out for a modest 212 after winning the toss, and Root's 100 not out took the hosts and tournament favourites to their target with more than 16 overs to spare. Big-hitting opener Jason Roy strained his left hamstring in the eighth over and didn't return to the field. Gayle went on to set the career record for most runs in ODIs between West Indies and England before he was finally caught on the boundary for 36.
There was the bonus, too, of 40 from Woakes, promoted to three in the reshuffle, before he was dismissed by a rare piece of good fielding from substitute Fabian Allen, who turned and gestured to the Southampton supporters who had tormented him.
England eased through their run chase, though Bairstow endured a scary moment when he was smashed on the grille by a delivery from Russell - Bairstow was fit to continue but Russell, who had returned for West Indies after missing Monday's washout against South Africa with a knee issue, had to be helped from the field after falling in his follow through. West Indies paceman Oshane Thomas and Sheldon Cottrell would love the extra bounce on this pitch.
When Nicholas Pooran and Shimron Hetmyer, the future of Caribbean batting, were together, West Indies could have tested England but Morgan turned to a seventh bowler in Root in the absence of a second spinner in Moeen and was instantly rewarded.
England were not flawless and coach Trevor Bayliss will not be happy with the two dropped catches by Mark Wood and Chris Woakes that could have proved costly if Chris Gayle and Andre Russell had been at their ebullient best.
Two balls after his dismissal West Indies were in deep trouble at 55/3 when Shai Hope was trapped in front of the wicket by Wood.
Top-ranked England entered the game with two wins and a loss, and West Indies had a win, a loss and a point from a washed out match against South Africa.