Asked to bowl first, SRH bowlers delivered in the crucial match, restricting the rivals to a modest 138 for seven.
The total proved too small for the defending champions as they overhauled the target with 10 balls to spare as opener Shikhar Dhawan (62) led the way with an unbeaten half-century.
SRH lost their captain David Warner (6) early but Dhawan and Moises Henriques (44) took the match away from MI with their fiery knocks.
Dhawan and Henriques added 91 runs for the second wicket in 11 overs before Jasprit Bumrah dismissed the Australian in the 13th over.
Dhawan's 62 came off 46 balls with six boundaries, including two sixes.
The left-hander finished the match in the company of Vijay Shankar (15) after cheap dismissal of Yuvraj Singh (9).
SRH remained at fourth position but now have 15 points, five more than closest rival Kings XI Punjab.
One more win, taking their tally to 17, will seal them a berth in the Play-offs because even if Punjab win their remaining three matches they can get maximum to 16 points.
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First Published: May 8, 2017, 11:53 PM IST