Saini made a memorable international debut by grabbing two important wickets off consecutive deliveries to help India restrict the West Indies to 95/9 in 20 overs after skipper Virat Kohli asked the Windies to bat first.
Coming to bowl in the fifth over after Washington Sundar and Bhuvneshwar Kumar had removed openers John Campbell and Evin Lewis for a pair of ducks, Saini (3/17) snared the wickets of Nicholas Pooran (20) and Shimron Hetmyer (0) off back-to-back deliveries to reduce the hosts to 28/4 after five overs.
West Indies could never recover from there even with comeback man Kieron Pollard top-scoring with a run-a-ball 49 (4x2, 6x4) before Saini got his back in the final over which he finished with a wicket maiden to cap off a dream debut.
Chasing 96 for victory, the visitors lost Shikhar Dhawan early, plumb in front to Sheldon Cottrell (2/20) in just the second over.
Skipper Kohli joined Rohit Sharma in the middle, the pair recently in the news for their reported rift after the World Cup.
Just when they were looking good, especially Rohit with some clinical hitting, the Indian vice-captain threw away his wicket for 24 from 25 balls (2x4, 2x6), trying to go for a needless big shot and holing out to Kieron Pollard at long-on off Sunil Narine (2/14).
Rishabh Pant showed little maturity, sweeping to Cottrell at deep square leg off Sunil Narine and getting out for a first ball duck leaving India in a spot of bother at 3/32.
Kohli held one end up as Manish Pandey (19) could not convert his good start, Keemo Paul (2/23) sending his stumps cartwheeling with a quick delivery.
It became tense when Kohli (19) top-edged a pull to Pollard at midwicket off Cottrell in the 14th over, with India needing 24 to win from 36 balls and five wickets in hand.
But Krunal Pandya (12) and Ravindra Jadeja (10 not out) guided the Men in Blue over the line with 16 balls to spare, as India took a 1-0 lead in the three match T20 series.
India scored 98/6 in 17.2 overs.
Earlier, it was all about Saini as he bowled with raw pace and in good lengths.
Besides Saini, pace spearhead Bhuvneshwar returned figures of 2/19 while Sundar (1/18), Khaleel Ahmed (1/8), Krunal Pandya (1/20) and Ravindra Jadeja (1/13) picked up one wicket each.
It all started for 26-year old Saini, blessed with raw pace, when he found Pooran's top edge with a short of length delivery, which flew to wicketkeeper Rishabh Pant.
In the very next ball of the fifth over, Saini was on a hat-trick on debut after castling Shimron Hetmyer. Returning in the final over, Saini trapped the big-hitting Pollard in front, with reviews confirming the dismissal.
The Windies' batters were all over the place as apart from Pollard and Pooran, none scored in double digits.
Brief scores: West Indies: 95/9 in 20 overs (Kieron Pollard 49, Navdeep Saini 3/17) vs India 98/6 in 17.2 overs (Rohit Sharma 24, Sheldon Cottrell 2/20, Sunil Narine 2/14, Keemo Paul 2/23)
( With inputs from IANS )