Pakistan needed six runs from the final over to win, but the game was back on the line with Shahid Afridi out for 46 just moments earlier and two nervous tailenders yet to find any rhythm facing the hosts' attack.
But Babar hit Marlon Samuels' first delivery of that final over through the covers for four and, after two dot balls followed, the scores were tied on a leg bye with two deliveries remaining as the rain began to fall.
There was further drama when Saeed Ajmal was run out on the next ball, but with one run needed from the final delivery, Babar delivered it in style with a towering shot, hitting a low full toss over long-off.
Chasing the hosts' 152 for seven, Pakistan were ahead of the run rate but in danger of undoing themselves with needless dismissals, reduced to 10 for two early on and then losing Umar Akmal to a run-out and seeing Umar Amin stumped when everything appeared under control.
Afridi got things back on track, setting the tone for an innings worth 46 off 27 when he hit the first ball he faced for six, but the match was back in the balance when he was dismissed in the penultimate over.
Shannon Gabriel was the bowler, handed the ball at a critical point despite having given up 34 off his previous three overs, and it started badly for him when he gave up eight runs off the first three balls, four of them extras.
But he rewarded his captain's faith with the key wicket of Afridi to give the West Indies hope.
Babar, making his debut at the age of 34, had been living dangerously as he made just three from the first 12 balls faced, but when it mattered he began the final over with a four and ended it with the six that sealed the win.
Earlier, the West Indies had struggled against Pakistan's spinners during a rain-interrupted innings, needing an unbeaten 49 from Kieron Pollard to make 152.
- Sports & Recreation
- Shahid Afridi
- Marlon Samuels