Dhawan (92) and Rishabh (58) scored superb fifties to guide the team home in the last ball of the thrilling match.
India crushed West Indies by 71 runs in the second T20I played at the Atal Bihari Vajpayee Ekanta Stadium in Lucknow.
High-flying India overpowered West Indies in the first T20I of the three-match series at the Eden Gardens on Sunday.
Depleted West Indies search for redemption in natural habitat.
West Indies bowled out lowest scores against India in ODIs. India won series by 3-1.
West Indies eye top-order stability in bid to square series in the fifth ODI in Thiruvananthapuram on Thursday.
A dominant India thrashed the West Indies by 224 runs in the fourth ODI to take a 2-1 lead in the five-match series in Mumbai on Monday.
With the series levelled at 1-1, plenty will depend on India under-fire middle order if they are to regain the lead.
Virat Kohli scored his third consecutive century in series, but the visitors clinched the match by 43 runs in Pune on Saturday.
Kohli rested for three match home series against Windies while he lead in Australia.