But Suryakumar Yadav stood firm and smashed 34 not out off 18 deliveries to take the team home with Venkatesh Iyer, who hit the winning six in his unbeaten, 13-ball 24.
Earlier, Rohit took the attack to the opposition bowlers as he hit four fours and three sixes but mistimed a hit to the deep mid-wicket fielder where Odean Smith took a good catch.
The left-handed Kishan, who was bought for a whopping $2 million by Mumbai Indians in the IPL auction last Saturday, struggled on his way to 35 before falling to Chase.
Virat Kohli's poor run extended into the T20I format as he fell to Fabian Allen for 17. He had scores of eight, 18 and nought in India's ODI clean-sweep of the West Indies.
Sheldon Cottrell's wicket of Rishabh Pant raised the tension in the Indian dug-out, but Yadav and Iyer put on an unbeaten stand of 48.
Put into bat, Pooran built key partnerships including a 45-run sixth-wicket stand with returning skipper Kieron Pollard, who hit an unbeaten 24 after he missed the last two ODIs due to a knee injury.
Pace bowler Bhuvneshwar Kumar dismissed opener Brandon King for four with the fifth ball of the match before Kyle Mayers and Pooran, who hit four fours and five sixes in his 43-ball blitz, attempted to rebuild the innings.
The left-handed Mayers made 31 off 24 deliveries but fell lbw to Yuzvendra Chahal's leg-spin while Bishnoi slowed the flow of runs from the other end.
Bishnoi, 21, trapped Chase lbw for four and dismissed Rovman Powell cheaply three balls later.
Pooran, who was given an early reprieve on eight when Bishnoi took a catch in the deep but his foot touched the boundary rope, went on to complete his sixth T20 international fifty.
The second match is on Friday, with all the games to be played at Kolkata's Eden Gardens.