Salt led an improbable English fightback with 109 not out while middle order batsman Harry Brook also emerged as a hero with an unbeaten 31 as England finished on 226-3 with a ball to spare to eclipse the West Indies' mammoth total of 222-6.
Brook's blistering late knock included 24 off the final over, the 24-year-old Yorkshireman sealing a remarkable victory with a six off West Indies bowler Andre Russell over third man.
The win leaves the best-of-five series finely poised, with West Indies leading 2-1 with two games to play.
Man-of-the-match Salt's century from 56 balls included four fours and nine sixes, while Brook's unbeaten 31 came off just seven balls and included four sixes.
Set a target of 223 to win, England made a rollicking start to their run chase, with Salt aided by 51 from skipper Jos Buttler.
But a tight spell of bowling from the West Indies and the dismissal of Buttler staunched the flow of English runs and the hosts looked to be on course for victory as the run rate crept ever higher.
Yet England yanked the initiative back spectacularly, with Liam Livingstone hitting 30 off 18 balls before Brook's remarkable late cameo dragged the tourists over the line.
"It feels like it's been a long time coming in an England shirt," Salt said of his century. "It's special to do it today with the series on the line. Brooky came in and played a hell of a knock at the end."
Earlier, West Indies appeared well set to clinch the series after Nicholas Pooran smashed 82 from 45 balls, following a shaky start from the home side.
The West Indian wicketkeeper's superb knock came after England had the hosts reeling at 8-2 with the early dismissals of openers Brandon King and Kyle Mayers.
Pooran's knock was the centerpiece of a sustained middle order onslaught with Rovman Powell adding 39, Sherfane Rutherford 29 and Shai Hope 26.
The hosts devoured England's bowling, putting on a breathtaking exhibition of power hitting that yielded more sixes (16) than boundaries (14).
England had made two changes to the side beaten by 10 runs in the second T20 on Thursday, bringing in Reece Topley and Gus Atkinson for Rehan Ahmed and Chris Woakes.
But both Topley and Atkinson struggled to make much impression with the ball.
Topley made an early breakthrough to remove Mayers for a duck but conceded 32 runs from his four overs at eight an over. Atkinson meanwhile was battered for 33 runs off two overs.
Game four of the series takes place in Trinidad & Tobago on Tuesday.
Brief scores:
West Indies 222-6 (N. Pooran 82, R. Powell 39, S. Rutherford 29; M. Ali 1-26, A. Rashid 2-32)
England 226-3 (19.5 overs) (P. Salt 109 not out, J. Buttler 51, H. Brook 31 not out; G. Motie 1-30, J. Holder 1-52)
Result: England won by seven wickets
Series: West Indies lead best-of-five series 2-1
