West Indies lost despite their best batting performance of the series, with opener Evin Lewis producing a brilliant 176 from 130 balls before he had to leave on a stretcher, retiring hurt.
Moeen Ali and Jos Buttler launched a stunning counter-attack. The duo added an unbeaten 77 runs in just 48 balls that took England just past the DLS score.
Three uncapped players - paceman Craig Overton, legspinner Mason Crane and wicketkeeper Ben Foakes - were called up to the squad that was announced on Wednesday.
Ali bludgeoned 102 off 57 balls, with seven fours and eight sixes, to get England to 369/9 batting first, their fifth-highest ODI total in the history.
England are in dire need of batsmen like Buttler or Hales to support England's Test skipper Joe Root and veteran batsman, Alastair Cook.
Jones could be a doubt for the Ashes series in Australia after sustaining a stress fracture in his lower back.
Bairstow was 100 not out, reaching three figures on the penultimate ball of the match. Next delivery, Ben Stokes launched a six over long-on to seal the win.
Chandigarh's left-handed batsman scripted history on this day as India walloped England with his six sixes.
Gayle's innings on Saturday against England during the one-off T20I was a reminder for the cricket fraternity across the world that the 37-year-old Jamaican has lost none of his power at the crease.
In the list of hitting the most number of maximums in T20Is, the sturdy Jamaican is followed by New Zealand’s Brendon McCullum (91 sixes) and Australia’s Shane Watson (83 sixes).
Lewis and Gayle smashed 72 runs off 39 balls for the opening wicket before West Indies lost its momentum. Lewis made 51 off 28 balls and Gayle 40 off 21 balls.
On day three, ultimately the final day, England took control through Anderson's 7-42, the best innings return for the veteran seamer in his 129th test.
West Indies emerged with a 22-run lead against England after a rain-affected day which kept conditions ideal for the quick bowlers at Lord's.
All-rounder Ben Stokes took 6/22 for England in a sizzling spell of swing bowling to dent West Indies' hopes of a first series win in England since 1988 and the era of Viv Richards.
Bangladesh, which had a notorious run of 71 defeats in 72 internationals from 1999 to 2004, have transformed into a team capable of surprising many and causing upsets.
By beating its nearest challenger for the one automatic qualification spot, England moved five points clear in Group F of the European zone with two rounds remaining.
Kane got off the mark for this season, scoring the first and last of England's four second-half goals against the world's 190th-ranked team in Group F of European qualifying.
Nurse has been named as a replacement for leg-spinner Samuel Badree, who was unavailable due to a prior commitment.
Rooney last month called time on his often tempestuous international career, retiring from duty after spurning an offer to return to the squad.
West Indies last won a test in England 17 years ago.
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