Mumbai last won the trophy way back in 2006-07 when they defeated Rajasthan.
Opting to bowl after winning the toss, pacers, Shivam Dubey (3/29) and Dhawal Kulkarni (3/30) picked up three wickets apiece as Mumbai bowled out Delhi for a paltry 177 in 45.4 overs.
Later, Tare starred with the bat with an 89-ball 71 that included 13 boundaries and a six and in the company of crisis man Siddesh Lad (48) stitched 105 runs for the fifth wicket to seal the win for Mumbai with 15 overs to spare.
But Tare had his heart in his mouth in 19th over of the innings when he survived a caught behind decision. Repeated TV replays showed the ball had bounced before nestling into Delhi wicket-keeper Unmukt Chand's gloves.
Tare eventually fell prey to Manan Sharma, trapped plumb in front of the wicket in the 31st over but by then he had already taken Mumbai to sniffing distance of a win.
Lad's valiant 48 came off 68 deliveries with four boundaries and two sixes but he fell two short of his half century, caught by Dhruv Shorey off Lalit Yadav with Mumbai scoreboard reading 176 for six.
Lad also survived a run out appeal at the non-strikers end after the umpires could not come to a conclusive proof to suggest that the straight drive off Tare hit the boots of Subodh Bhati and ricocheted onto the stumps.
The benefit of doubt went in favour of the batsman.
Chasing 178, Mumbai lost in-form Prithvi Shaw for eight in third ball of the innings after he smashed two consecutive boundaries of Navdeep Saini.
Ajinkya Rahane (10), captain Shreyas Iyer (7) and Suryakumar Yadav (4) all departed cheaply to Saini and Kulwant Khejrolia, leaving Mumbai struggling at 40 for four in 7.4 overs.
Earlier, Delhi began their innings on a bad note with Mumbai pacers Kulkarni and Tushar Deshpande exploiting the moist conditions.
Deshpande (2/30), Dubey and Kulkarni picked up eight wickets between them to rattle Delhi's batting line-up.
Captain Gautam Gambhir got out for one in second over off the innings. Soon Unmukt Chand (13) and Manan Sharma (5) also went cheaply.
Delhi were reeling at 21 for three in six overs, but Nitish Rana (13) and Dhruv Shorey (31) tried to stabilise the innings for a brief period before the former was removed by Dubey, leaving Delhi precariously placed at 60 for four.
Shorey too did not last long as he was stumped by Tare off a brilliant Shams Mulani's delivery.
Thereafter, Pawan Negi (21), Subodh Bhati (25) and Himmat Singh (41) scored vital runs to take Delhi to 177.
But once Negi walked back to the pavilion retired hurt, Mumbai ran through Delhi's tail in quick time.