It was the first five-wicket haul of this season and best bowling figures for an Australian in T20s as Zampa wrecked havoc on the Sunrisers in the slog overs, claiming six for 19 in his stipulated 4 overs.
The 24-year-old Australian 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.
Earlier electing to bat first, Sunrisers found the going tough from the start as 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.
Other opener Shikhar Dhawan (33) struck a couple of fours and sixes before holing out to Sourabh Tiwary off R Ashwin at long-on as Hyderabad slipped to 64 for 2 in the 11th over.
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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.
Zampa picked up Moises Henrique (10) in the next delivery with Bhatia taking the catch once again as Hyderabad slumped to 114-5 in 18 overs.
In a thrilling last over, Zampa was sent over mid-wicket in the first ball by Hooda but after a brief chat with skipper Mahendra Singh 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.