Powered by David Warner's fighting 65-run knock and his stellar 111-run partnership with Lokesh Rahul (46), Hyderabad overcame a shaky start to score a competitive 172 for five after being invited to bat.
After losing three quick wickets, Mumbai fought through big-hitting Kieron Pollard (78) and Ambati Rayudu (35 off 27) but all they could manage was 157 for seven in the end.
Pollard prevented it from becoming a one-sided match with his scintillating 48-ball knock, embellished with six sixes and three fours.
Irfan Pathan bowled Pollard in the first ball of the final over, in which Mubai needed 20 runs. It was Pollard, who had brought Mumbai this far and his wicket at the crucial juncture dashed hopes of the defending champions.
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An intimidating Steyn had Corey Anderson (1) caught on a well-directed bouncer. Darren Sammy joined the party by castling opener Ben Dunk (20), reducing Mumbai to 31 for 3 inside powerplay overs.
Pollard and Rayudu brought some respectability to the Mumbai's chase with their fighting knocks. The tall West Indian hit spinner Amit Mishra for two sixes and a four in the 13th over and dispatched Karan Sharma for a humongous hit in the next over to raise hopes of a turnaround.
Pathan broke the stand by dismissing Rayudu but there was no stopping Pollard, who again took Mishra (0/54) to cleaners by hitting him for three consecutive sixes and a four in the 17 over.
Earlier, Warner worked hard for runs initially but once he got his eye in and was settled, the Australian left-hander scored runs at a brisk pace. He hit more sixes (4) than fours (1) in his 51-ball knock before departing towards the end of the SRH innings.