Winning his first toss after a four-month injury absence, Buttler put the hosts in to bat first, with big-hitting lower order cameos from Gudakesh Motie and Romario Shepherd helping West Indies set an imposing 183 to win.
Salt's super unbeaten innings of 103, which included six sixes, was backed up by Jacob Bethell's maiden T20I half-century, after Buttler had fallen for a golden duck, to see England record the highest-ever successful T20I run chase at Kensington Oval.
After beating England 2-1 in the one-day series, West Indies will hope to complete a T20 turnaround, with the second of five matches scheduled for Sunday back in Bridgetown.
With fast bowler Jofra Archer rested after playing all three of England's ODIs in the Caribbean, Saqib Mahmood came into the bowling attack and showed no signs of rust as he struck three times in four overs to leave West Indies languishing on 3-18.
The hosts continued to be ultra-aggressive, finding the boundary with regularity, but the wickets also kept on falling. When Adil Rashid had Akeal Hosein trapped leg before wicket, England were firmly in the driving seat.
A handy 49-run, ninth-wicket partnership between Shepherd and Motie, the latter smashing a thrilling 33 off 14 balls, ensured England would have to produce a Kensington Oval first to get their T20 series off to a winning start.
Salt, on the back of his slowest-ever ODI half-century in England's defeat in the one-day series decider on Wednesday, was not hanging around this time, racing to 50 from 25 balls inside the first powerplay.
Will Jacks' departure for 17 brought Buttler to the crease, but the skipper was soon heading back to the pavilion as Motie took a stunning one-handed catch on the boundary.
Without Alzarri Joseph in the bowling attack, the fast bowler suspended for two matches for storming off the field after a row with captain Shai Hope during Wednesday's win, the hosts struggled to trouble Salt and Bethell.
Salt moved past 1000 T20I runs as he kept finding the boundary, with a driven four down the ground taking him to his third century in the format, before Bethell clubbed the winning runs, sealing England's victory with three overs to spare.
