Conway put New Zealand back on course with a 52-run partnership with Martin Guptill who made 35 from 27 balls with three fours and two sixes.
South Africa-born Conway made 126, including a century from 95 balls, and shared a record 159-run partnership with Mitchell as New Zealand made 318-6 batting first after winning the toss.
Set a target of 143 for victory in the fifth and final match, New Zealand claimed victory by seven wickets and took the series 3-2 with an emphatic 27 balls to spare.
Conway's base price in the IPL auctions was Rs 50 lakh.
The left-handed batsman's previous four T20 scores were - 93*, 91*, 69, 50 in the Super Smash.
New Zealand struggled early against Australia pacemen Daniel Sams, Jhye Richardson and Kane Richardson before Conway led a concerted rival.