Punjab won the toss and elected to bowl before the rain arrived, although no overs were lost.
When play eventually started, Australia all-rounder Marcus Stoinis removed India opener Rohit Sharma for just eight, before Varma and Bairstow combined for 51 runs.
Another strong partnership worth 72 between Varma and Suryakumar put Mumbai in a decent position before both fell in the space of three deliveries.
However, Naman Dhir’s 37 off 18 balls boosted the five-time champions to a challenging total.
In reply, Australia wicketkeeper Josh Inglis sparked Punjab with 38 off 21, including 20 off one over from India fast bowler Jasprit Bumrah.
But Mumbai hit back as captain Hardik Pandya removed Inglis and were marginal favourites when Nehal Wadhera fell for 48, with Kings needing 48 off 26 balls.
However, Shreyas kept his side in contention and, requiring 23 runs off the last two overs, he flicked the switch to smash four sixes in the 19th over in a clinical finish.
The final is a repeat of the first qualifier match, in which RCB thrashed Kings by eight wickets.
Both sides scored 19 points in the group stage, with Kings narrowly topping the table on net-run rate.
Punjab were beaten by Kolkata Knight Riders in their only previous final appearance in 2014, while RCB have lost the showpiece in 2009, 2011 and 2016.
