With two runs needed from the final delivery, Bhuvneshwar (3/41) trapped Rovman Powell (27) lbw with a full toss, leaving RR at 200 for seven while chasing a target of 202.
After electing to bat, Sunrisers Hyderabad posted 201 for three thanks to explosive fifties from Nitish Reddy (76 not out) and Travis Head (58), along with Heinrich Klassen's unbeaten 42 off 19 balls.
Things were looking up when Bhuvneshwar removed Jos Buttler and Sanju Samson for ducks in the opener to leave RR at 1 for 2.
However, young people Yashasvi Jaiswal (67) and Riyan Parag (77) counter-attacked the fifties en route to a massive 134-run partnership to lay the foundation for the running game.
But once the duo returned to the hut, Shimron Hetmyar (13) hit a four and a six as it looked like a regulation chase, but T Natarajan got rid of Hetmyar, who perished in the long term and then captain Patrick Cummins removed Dhruv Jurel, who was caught at deep square leg in the 19th over.
The sudden loss of wickets gave SRH hope of a turnaround, but Powell produced a six in the last ball of the 19th over to bring the equation down to 13 off six balls.
The West Indian kept his head calm to bring the score to 2 off the last ball but Bhuvneshwar denied him the glory.
After the win, SRH ended their two-game losing streak and moved to fourth place with 12 points, as did Kolkata Knight Riders, Lucknow Super Giants, while RR remained at the top with 16 points.
Chasing the total, Jaiswal and Parag were dropped early in their innings and they used the reprieve to plan a quick recovery.
Parag broke the chains in the third over when he carried Bhuvneshwar for two fours and a six. Jaiswal then blasted Marco Jansen for two boundaries after Cummins bowled a direct catch.
Parag, who was dropped on 24 by Abhishek Sharma, hammered Cummins for another maximum, before carrying Jansen for two more fours.
Cummins finally introduced some spin into the attack, but Jaiswal smashed Shahbaz Ahmed for a four to complete his fifty. Three balls later, Parag completed a reverse sweep four to reach his milestone in 32 balls.
Jaiwal then played a T Natarajan delivery to the stumps, while two overs later Parag was run out at long-on.
Earlier, Head forged a 96-run stand from 57 balls for the third wicket with Nitish, who also added another 70 from 32 balls with Klassen to take SRH beyond the 200-mark for the fifth time this season.
Head, who was dropped by Parag off the first ball of the innings, hit six fours and three sixes, while Nitish smashed eight maximums and three fours during his brutal assault.
Playing his 300th match, Yuzvendra Chahal, who was selected for the T20 World Cup, was on the firing line as he conceded 62 in his quota of four overs.
Avesh Khan (2/39) made the first breakthrough, having Abhishek (12) caught at deep mid-wicket. Sandeep Sharma (1/31) then took a first-ball wicket, removing Anmolpreet Singh (5) as SRH scored 37 for 2 in the powerplay, their lowest this IPL.
The spin duo of Chahal and R Ashwin (0/36), however, disappointed as the batters milked them for runs to put SRH on the path to recovery.
Chahal had a horrible night as Head smashed him for two sixes and a four during an 18-run stand after he came in to bowl immediately after the powerplay.
The leg-spinner was then taken down by Nitish, who plundered him for two sixes and two fours to pile up 21 runs.
Nitish, who also hit Ashwin and Avesh for two maximums, then sent Sandeep sailing under extra cover, while Head pulled Avesh for six more in the 15th over.
The Australian, however, played lead role on his middle stump as SRH slipped to 131/3.
New man Klassen then joined the party, sending Chahal into orbit with consecutive sixes. When Trent Boult was back, Klassen picked him for two fours, while taking another six off Sandeep.
(With inputs from PTI)