England excused four of their first-choice side after the drawn ODI series - Jos Buttler, Jason Roy, Ben Stokes and Moeen Ali all watching from home - but Bairstow plugged a big gap in the batting line-up with a career-best 68 from 40 balls.
Opening for the first time in 28 matches he broke the back of a 161-run target, which owed much to Rashid's miserly return of one for 15. Tom Curran walked away with four wickets as the Windies posted 160 for eight but it was Rashid's precise stint which did most to stall the home side.
Joe Denly and Sam Billings both fell with the end line in sight but Curran reappeared to hit the winning boundary with seven balls remaining.
Curran made short work of Shai Hope with the new ball but the real priority was seeing off Chris Gayle. Man-of-the-series in the ODIs, clubbing 39 sixes along the way, he soaked up five dot balls off David Willey before clearing the ropes at third man and wide long-on. That was the height of his success this time, though.
Chris Jordan was his downfall, summoning a wide yorker which Gayle could only squeeze to short third man. Shimron Hetmyer also failed to leave his mark on proceedings, chipping in with 14 before pumping a Curran full toss straight to mid-on.
At the halfway mark England had restricted their hosts to a gentle 67 for three, with the Windies unable to score off 29 of the 60 legal deliveries.
Darren Bravo and Nicholas Pooran helped claw back some of the lost ground, Liam Plunkett suffering worst as he sent down four wicketless overs for 44. The fourth-wicket pair added 64 in 51 balls but were parted in style by the returning Chris Jordan.
He confounded Bravo with a disguised slower ball and leapt into action when the ball popped back towards him, throwing himself one-handed in front of the non-striker.
Curran was nevertheless favoured for the death overs and claimed two more victims, cleaning up Pooran for 58 and Jason Holder with another full toss.
The reply began in a flurry of activity as Sheldon Cottrell's opening over contained 17 runs, a dropped catch and finally the wicket of Alex Hales, heaving hard and losing his stumps.
Joe Root's hopes of sharpening his T20 skills lasted exactly two balls, lbw to Cottrell attempting to open up the leg-side, but Bairstow was unaffected.
He launched himself into a 13-ball sequence containing five boundaries and one six, driving England to 62 for two in the powerplay, well ahead of the Windies benchmark.
Bairstow nicked Carlos Brathwaite on 34 but a fingertip catch skimmed off Hope's gloves and brought him four more. Brathwaite did manage to see off skipper Eoin Morgan but Bairstow was dropped again by Cottrell shortly after reaching a 27-ball 50 with his second six.
By the time he was held at wide long-on, Brathwaite the catcher off Ashley Nurse's off-spin, he had left a target of 58 in 50 balls.
Billings and Denly, both of whom are still seeking to find their place in the England set-up, chalked off all but eight of the required runs in calm fashion before nerves struck.
Holder persuaded Denly to hole out with 14 balls remaining then Billings had an almighty let-off next ball, Oshane Thomas just failing to hang to a mis-cued pull at fine leg.
Reprieved, Billings then swiped at a Cottrell cutter that knocked back his stumps. Four byes over Hope's head eased the growing tension allowing Curran to slog the winning runs.
Join our new commenting forum
Join thought-provoking conversations, follow other Independent readers and see their replies