Miller played a scintillating 41-ball 80 knock laced with five fours and an equal number of sixes while Mandeep hit 58-ball 77, decorated with seven fours, to help their team secure victory with just one ball to spare.
Needing 29 off last two overs, Miller hit two consecutive fours off Dinda in the 19th over to ease down the pressure.
Kings XI managed to get 13 runs off the over.
With 16 required off the last over, Miller hit two huge sixes off Wright to hand the hosts their third win from six matches. Both added an unconquered 128 runs in just 79 deliveries for the fourth-wicket partnership to script an impressive victory for the Punjab team.
Pune rode on skipper Aaron Finch's strokeful half-century and significant contributions from Yuvraj Singh and Luke Wright to post an imposing 185 for four.
Finch scored an attractive 42-ball 65 with the help of eight fours and two sixes while Yuvraj hit an impressive 24-ball 34, laced with two fours and three sixes. Wright helped the visitors finish with flourish, courtesy his breezy 34 off 10, during which he found the fence six times.
In fact, 17th over from Azhar Mahmood yielded 17 runs as Wright hit him for four consecutive boundaries.
Opener Robin Uthappa also scored a useful 37 off 33 deliveries with the help of four fours and added 83 runs for the opening wicket stand with Finch.
Chasing the target, Kings XI got off to a disastrous start as they were two down with just five runs on board.
While Bhuvneshwar Kumar dismissed Gilchrist in the first over, Ajantha Mendis struck in the second over to send back Azhar Mahmood putting Punjab on back foot.
Gilchrist scored just a boundary before getting caught behind by Robin Uthappa. Mahmood failed to get off the mark and was trapped leg before.