With two runs needed off the last delivery, Bhuvneshwar (3/41) trapped Rovman Powell (27lb) with a full delivery to leave RR with 200 for seven while chasing a target of 202.
Opting to bat, Sunrisers Hyderabad posted a score of 201 for three thanks to explosive 50s from Nitish Reddy (76 not out) and Travis Head (58), as well as Heinrich Klassen’s unbeaten 42 off 19 balls.
Things started to look up when Bhuvneshwar removed Jos Buttler and Sanju Samson for a duck in the opening over, giving RR a 1-2 lead.
However, teenagers Yashasvi Jaiswal (67) and Riyan Parag (77) put up a counter-attacking 50 in a massive partnership of 134 runs to set the stage for runs.
But as the duo returned to the hut, Shimron Hetmyar (13) hit a four and a six as it looked like a routine chase, but T Natarajan Get rid of Hetmyar, he died for a long time, then the captain Pat Cummins Dhruv Jurel was dismissed, who fell to deep square leg in the 19th over.
The sudden loss of wickets gave SRH hope of turning things around, but Powell’s six off the last ball of the 19th over reduced the equation to 13 off six balls.
The West Indies kept their cool and took the score to 2 off the last ball, but Bhuvneshwar was denied the glory.
After this victory, SRH ended their two-match losing streak and moved up to fourth place with 12 points, tied with Kolkata Knight Riders and Lucknow Supergiants, while RR topped the list with 16 points.
Chasing the total, Jaiswal and Parag were dropped early in the innings and they used the reprieve to recover quickly.
Parag broke the mold in the third over as he pushed Bhuvneshwar for two fours and a six. After a direct catch from Cummins, Jaiswal blasted Marco Jansen for two boundaries.
Parag was dismissed by Abhishek Sharma in the 24th over and he again top-scored off Cummins before hitting Jansen for two fours.
Cummins eventually introduced spin in the attack but Jaiswal smashed Shahbaz Ahmed for four to complete his fifty. Three balls later, Parag completed a reverse sweep of four balls to reach the 32-ball milestone.
Jawar then bowled T Natarajan to the stumps, while Parag was bowled out at long-on two overs later.
Earlier, Hyde had put up a stand of 96 off 57 balls with Nitish for the third wicket, who also added another 70 off 32 balls with Klassen to lead SRH to their fifth breakthrough of the season 200 points mark.
Hyder, bowled by Parag on the first ball of the innings, hit six fours and three sixes, while Nitish smashed eight maximum balls and three sixes in his brutal attack Four points.
Yuzvendra Chahal, who was selected for the T20 World Cup and was playing his 300th match, was in the firing line after conceding 62 balls in four overs.
Avesh Khan (2/39) got the first breakthrough when Abhishek (12) was caught deep at mid-wicket. Sandeep Sharma (1/31) then took the first wicket off Anmolpreet Singh (5) as SRH were in the powerplay 37 runs for 2, their lowest score in this IPL.
However, the spin duo of Chahal and R Ashwin (0/36) disappointed as the batsmen milked them for runs, putting SRH on the road to recovery.
Chahal had a terrible night, with Hyde bowling him for 18, hitting two sixes and a four after coming in to bowl immediately after the powerplay.
The leg-spinner was then bowled by Nitish, who hit him with two sixes and two fours to amass 21 runs.
Nitish also bowled Ashwin and Avish for two maximum runs before sending Sandeep over extra cover, while Hyde bowled Avish for another six in the 15th over.
However, the Australian hit the pacer to his middle stump and SRH slipped to 131/3.
Debutant Klassen then joined the team to send Chahal to orbit for six consecutive overs. When Trent Boult returned, Klassen picked him for two fours while also hitting a six off Sandeep.
(Inputs from PTI)