The 23-year-old opening batsman hit 101 from 132 balls as Pakistan reached their target of 228 for the loss of four wickets.
Younis Khan made 58, putting on a third-wicket stand of 112 with Jamshed, while Shoaib Malik contributed an unbeaten 34, sealing victory with a four with 11 balls to spare.
India, playing their first match since the one-day retirement of Sachin Tendulkar, made 228 for six and the "Little Master" proved a predictably tough act to follow.
The top five scored a combined 18 and only a magnificent 113 not out from their captain MS Dhoni made their total remotely competitive as Pakistan went 1-0 up in the three-match series.
Junaid Khan did most of the damage to the top order and finished with four wickets for 43.
Pakistan's chase got off to the worst possible start though as they lost the wicket of Mohammad Hafeez first ball, Bhuvneshwar Kumar's delivery taking out the off stump as the batsman failed to offer a shot.
Jamshed said: "It was a tough start when we lost Hafeez off the first ball, but we played out the tough period and now we are confident in this series."
Dhoni, who was named man of the match for his century, said at the post-match presentation: "[It was a] bad toss to lose, but I am happy with my batting.
"There was something for the fast bowlers, but they also pitched it in the right areas. Once we lost five wickets, it was difficult to come back.
"We delayed the slog to get to a respectable score. I think 227 was a good score from where we were, but not enough in the end. The wicket got better later and our part-timers struggled."
His opposite number Misbah-ul-Haq singled out Junaid and Jamshed for special praise.
"The spell by Junaid Khan was very good," he said. "He has done it before in a Test match in Sri Lanka. Good to see that the bowlers pitched in the right areas in helping conditions."
He added: "Nasir played an extremely good knock and showed the way by finishing the game. Humidity was a factor as both teams found it hard."
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