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’s 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’ 152-7, Pakistan were reduced to 10-2 early on. 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 he was dismissed in the penultimate over off Shannon Gabriel, who had conceded 34 off his previous three overs.
Babar, making his debut at the age of 34, 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, relying chiefly on an unbeaten 49 from Kieron Pollard to reach 152.