Chasing a modest 128 for a win, the table-toppers made heavy weather of the run chase, achieving the target on the last delivery of the match.
The SRH bowlers did well to take the match thus far but Rahane's 56-ball 62 proved to be the difference between the two sides.
Man-of-the-match Rahane, who registered his first half-century of the this season, and Sanju Samson (26) worked well for their 64-run opening stand before a few wickets made the match exciting in the end.
But come the 11th over and Ravi Bopara was more than lucky as wicketkeeper Naman Ojha came up with a brilliant diving catch to send back Samson back in the dug out to give SRH some hope.
The wicket did dry up the flow of runs for the visitors, but the lack of runs on the board helped their cause as Rahane stayed on with skipper Steven Smith (13) to bring them closer to a win.
Smith though was done in by leg-spinner Karn Sharma in the 14th over and Bopara struck for the second time in the very next over to get Karun Nair (1) caught and bowled.
Rahane, whose knock was lace with 9 boundaries, couldn't enjoy taking the team to the finish line as Trent Boult got the top-scorer clean bowled in the 19th over.
Earlier, backing Smith's decision to field first on a slowish track, RR bowlers Dhawal Kulkarni and Pravin Tambe picked up two wickets each to restrict SRH to 127 for five in 20 overs.
