RCB first posted 173-6 from their 20 overs after winning the toss.
Wicket-keeper/batsman Anuj Rawat top scored with a 25-ball 48 while Dinesh Karthik and captain Faf Du Plessis provided good support with 38 and 35, respectively.
Mustafizur Rahman was excellent with the ball for the Super Kings with 4-29 from his four overs.
CSK then used contributions from all their batsmen to easily reach 176-4 with eight balls to spare.
Debutant Rachin Ravindra led the way with 37 while Shivam Dube (34*), Ajinkya Rahane (27) and Ravindra Jadeja (25*) also played well.
Cameron Green took 2-27 from his three overs for RCB.
Full Scores: Royal Challengers Bengaluru 173-6 off 20 overs (Anuj Rawat 48, Dinesh Karthik 38*, Mustafizur Rahman 4-29)
Chennai Super Kings 176-4 off 18.4 overs (Rachin Ravindra 37, Shivam Dube 34*, Ajinkya Rahane 27, Ravindra Jadeja 25*, Cameron Green 2-27)
An unbeaten opening partnership between Jaiswal (93 not out) and new captain Gill (58 not out) in Harare secured the triumph for the reigning T20 world champions with one match of the series remaining.
Having started impressively, Zimbabwe lost both Tadiwanashe Marumani (25) and Wessly Madhevere (32) in quick succession, which ultimately kickstarted their collapse.
Sikandar Raza attempted to steady the ship with a crisp 46 from 28 balls, but the wickets continued to tumble as Shivam Dube (1-11) and Khaleel Ahmed (2-32) helped restrict Zimbabwe to a manageable 152 -7.
India wasted no time in chasing down their target, with Jaiswal and Gill reaching 61 by the end of the powerplay.
Jaiswal completed his half-century from 29 deliveries with the help of nine fours, with Gill then sealing his second consecutive 50 against Zimbabwe.
The win was secured in the second ball of the 15th over, putting India 3-1 up in the series.
Data Debrief: India's dominance over Zimbabwe continues
India clinched the series with a third win after losing the first game, making it the third time in four bilateral men's T20I series against Zimbabwe that they have won (D1).
They have now won six of their last seven men's T20Is against Zimbabwe (L1).
Raza's knock for Zimbabwe saw him make history, becoming the first player from the nation to score 2,000 runs in T20Is.
Suryakumar Yadav hit an unbeaten half-century as India successfully chased down their opponents' target of 111 for a seven-wicket victory - and their third win in as many Group A matches.
The tournament co-hosts also entered the contest with a perfect record from their opening two games, but endured a poor start as Arshdeep Singh claimed Shayan Jahangir and Andries Gous in the first over.
That left them at 3-2, but they managed to settle with Steven Taylor (24 from 30 balls) and Nitish Kumar (27 off 23) top-scoring as their side finished at 110-8.
However, the States made a purposeful start in the field. With the second ball, Virat Kohli was gone for a duck as Gous caught him from Saurabh Netravalkar, who also dismissed Rohit Sharma for just three.
Suryakumar (50 off 49 not out) stepped in and held the fort alongside Shivam Dube (31 off 45 not out), though India were still in danger of a potential shock, needing 35 off the final 30 balls.
Nevertheless, an untimely five-run penalty was awarded to them as the USA took longer than a minute between overs for the third time.
That swung the pendulum firmly in India's favour. Successive boundaries from Suryakumar eased the pressure as they saw out what was eventually a comfortable win.
Data Debrief: India through, but Kohli's struggles continue
It was not as comfortable as India would have hoped for but, nevertheless, they managed to get the job done, with Suryakumar leading the charge.
However, the form of Kohli will be a cause for concern. His dismissal for a golden duck means he has now scored just five runs in his nation's opening three matches.
