It was the best bowling figures by an Australian in T20s as Zampa claimed six for 19 in his four overs to help Pune restrict Sunrisers for a modest 137 for 8 after skipper David Warner elected to bat.
However, the Supergiants could not script a successful runchase as Sunrisers dished out a disciplined bowling effort, led by the veteran seamer Ashish Nehra, who claimed three wickets, to restrict Pune to 133-8 in 20 overs.
With this win, Sunrisers reached the top of the points table with 14 points, while Pune sunk to the seventh spot with six points after their eight defeat in 11 matches.
Chasing 138 to win, Pune were dealt a severe blow only in the third ball of the innings when seamer Bhuvneshwar Kumar removed opener Ajinkya Rahane (0) with Naman Ojha taking the catch, in a maiden over.
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Rahane has been their top-scorer and his dismissal early on put the pressure on the top-order as Usman Khawaja too was run-out in fourth over by Hooda when the opener was trying to steal a second run.
Ashwin blasted a couple of fours in the next two overs but Bailey's miscued shot ended in the hands of Ashish Nehra as Pune slipped to 68-3.
Ashwin picked up another four in the 13th over but a brilliant catch by Ojha in the next over off Barinder Sran cut shot his innings.
Nehra then dismissed Saurabh Tiwary (9) in the 15th over after being hit for a boundary. It was a miscued pull shot which ended up as a simple catch for Dhawan.
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Earlier, Zampa dismissed Yuvraj Singh (23), Kane Williamson (32), Moises Henriques (10), Deepak Hooda (14), Naman Ojha (7) and Bhuvneshwar Kumar (1) -- all of them perishing in the final overs while trying to go for big shots.
Even as the 24-year-old Australian claimed the second-best figures in all IPLs, fellow spinner Aswin (1/16) and seamer R P Singh (1/23) also contributed with one wicket each.
Sunrisers found the going tough as R P Singh gave Pune a good start by removing the dangerous David Warner (11) early on when he had him caught behind in the 4th over.
Yuvraj Singh (23) tried to step us the run-rate with a 21-ball 23 studded with two sixes and a four but he was sent packing by Zampa when the left-hander attempted to muscle him out of the park only to be caught by Tiwary at Long-off.
In the 17th over, Kane Williamson brought up the hundred for his team with a single. The New Zealand skipper blasted three fours in his 37-ball innings before Rajat Bhatia took a running catch at long-off to end his innings. It was Zampa's second wicket.
In the 19th over, Ashok Dinda bled 15 runs as Hooda and Ojha blasted a six and a four to take Hyderabad to 129-5.
In a thrilling last over, Zampa was sent over mid-wicket in the first ball by Hooda but after a brief chat with skipper Dhoni, the Australian had Hooda stumped.
The leg-spinner then claimed his fifth wicket when Ojha was left cramped for room and deflected the ball onto the stumps. Kumar was then dismissed when he tried to play a slog shot only to be caught by Tiwary at long-on.