Abhishek smashed 69 off just 30 deliveries to propel India to 223 all out before Arshdeep Singh claimed three wickets as the visitors were restricted to 209-7 in their chase, giving the hosts a 2-1 advantage with two matches remaining.
The result also ended a run of successful chases during the tour, with India finally defending a total after the West Indies had levelled the series by chasing down 250 in Friday's second match.
Abhishek set the tone with an innings featuring six fours and five sixes, reaching his half-century from only 18 balls. He combined with Ishan Kishan, who contributed 37 off 18 deliveries, in a 108-run opening partnership after India were asked to bat first.
Sanju Samson fell for three, but Abhishek quickly took the attack to the West Indies bowlers, helping India race to 87-1 during the powerplay.
However, Roston Chase turned the momentum by dismissing Kishan and India captain Shreyas Iyer for one in the same over. Shamar Springer then removed Abhishek, leaving the hosts struggling to maintain their early scoring rate.
Tilak Varma added 28 while Nitish Kumar Reddy made 21, but India's progress was further disrupted by a series of costly dismissals. Axar Patel contributed 20 before Prince Yadav's unbeaten 13 off six balls provided a late boost to take the total beyond 220.
The hosts made an immediate impact with the ball, with Arshdeep removing West Indies captain Shai Hope for five. Hope had struck an unbeaten 102 in the previous match, but India's left-arm seamer ensured the visitors' chase began on a different note.
Shimron Hetmyer briefly threatened to turn the contest around, smashing 43 off 26 balls, including a productive over against Arshdeep that yielded 19 runs. However, Naman Dhir dismissed him shortly afterwards and continued to trouble the West Indies batting line-up.
Dhir also removed Rovman Powell for five before sending back opener Kamil Pooran, who made 42 off 27 deliveries, to leave the visitors at 105-4 in the 11th over.
India's bowlers maintained their control through the middle overs, with Axar Patel helping restrict the scoring after an expensive outing in the previous match. Dhir and Reddy also made valuable contributions as the West Indies struggled to build sustained partnerships.
Sherfane Rutherford kept the visitors' hopes alive with 32 off 20 balls, but Arshdeep bowled him in the 18th over to swing the contest firmly in India's favour.
The pace bowlers then executed their yorkers effectively, with Arshdeep and Prince Yadav ensuring the West Indies could not produce another late surge to complete the chase.
The fourth T20 international will take place in Hyderabad on Wednesday, with India now one victory away from securing the series.
India earned their highest-ever T20I run chase when they successfully chalked off New Zealand's 225/4 with four wickets remaining in a run-soaked encounter in Christchurch on Thursday night.
Darren Sammy believes West Indies have taken a significant step forward after holding India to a 2-2 T20I series draw, with the head coach encouraged by the return of his team's confidence and trademark swagger.
Less than eight overs were bowled in the fifth and final T20I between India and West Indies as heavy rain fell in Bengaluru.
India star seamer Jasprit Bumrah is doubtful to feature during the upcoming five-match ODI series against New Zealand in November despite being fully fit.
