Chasing 163 for victory, table-toppers Mumbai faltered against a disciplined Pune bowling attack to end on 142 for nine in 20 overs. Parthiv Patel top-scored with 52.
Put into bat first, Pune's batsmen were also at the receiving end of some inspired Mumbai bowling as they struggled to get quick runs on a seemingly slow track and end on 162-4.
Pune batsmen Manoj Tiwary (58) and Ajinkya Rahane (56) recorded gritty half-centuries to ensure a competitive target for the home team at Mumbai's Wankhede Stadium.
But it was 17-year-old Sunder who turned it round for Pune with his impressive figures of 3-16, making up for the absence of star all-rounder Ben Stokes.
Sunder, who has just played five first-class games for his south Indian state of Tamil Nadu, struck twice in his opening over including the prized scalp of opposition skipper Rohit Sharma (1).
The lanky off-spinner also got the dangerous Kieron Pollard (7), forcing the burly West Indian to flick a flattish delivery to Pune skipper Steve Smith at short mid-wicket.
Pune fast bowler Shardul Thakur also took three wickets.
Earlier there was an entertaining blitz from wicketkeeper-batsman Mahendra Singh Dhoni, whose 26-ball unbeaten 40 gave Pune the much-needed push in the final few overs.
Just when it looked like Pune's innings would end in a whimper, Tiwary and Dhoni combined to hit a flurry of fours and sixes in the final two overs that cost Mumbai 41 runs.
Mumbai have another shot at booking a final berth when they take on the winner of the eliminator between Sunrisers Hyderabad and Kolkata Knight Riders in the second qualifier on Friday.