Chasing 177, Dhoni scored an unbeaten 61 off just 34 balls with the help of five boundaries and three sixes, to steer the Pune to a thrilling victory in the last ball, reaching 179 for four in 20 overs.
Tripathi made a quickfire 41-ball 59 at the top of the order to give Pune a blazing start to their innings.
Earlier, Rising Pune Supergiant produced a disciplined bowling effort before a smashing half-century by Moises Henriques enabled Sunrisers Hyderabad to reach a competitive total of 176 for three.