Needing two off the final ball, Rajasthan's Rovman Powell missed a full toss and was trapped in front of the wicket by India international Bhuvneshwar Kumar.
The right-arm swing bowler, who finished with figures of 41-3, earlier undermined Rajasthan's innings by removing star England batter Jos Butler and captain Sanju Samson for nought in the first over in the first over the chase.
"I wasn't thinking much about the result in the last over," said Kumar.
"There was no discussion in the last over, was just focussed on the process."
After the early setbacks, young Indian batters Yashasvi Jaiswal and Rayan Parag scored counter-attacking half-centuries to rebuild the innings and Rajasthan seemed on course for a comfortable victory.
But Jaiswal (67 off 40 balls) and Parag (77 off 49) fell in quick succession to left-arm pacer T Natrajan, setting off a collapse.
Hyderabad captain Pat Cummins conceded only seven runs in a miserly penultimate to tilt the match in the home team's favour.
"Knowing the nature of the IPL, you never win the game until you actually win the game," said Samson.
In their innings, Rajasthan Royals made an uncharacteristically slow and shaky start with in-form opener Abhishek Sharma falling for 12 off 10 balls.
Anmolpreet Singh, who followed at number three, also failed to make an impact scoring a run-a-ball five.
But Australia international Travis Head and all-rounder Nitish Kumar Reddy upped the pace with a partnership of 96 runs off 47 balls.
The duo were particularly harsh on wrist spinner Yuzvendra Chahal who leaked 62 runs in his four overs.
Head's crucial knock of 57 off 44 balls finally ended when he chopped one onto his stumps as he tried to scoop pacer Avesh Khan.
South Africa's power hitter Heinrich Klaasen then joined Reddy who remained unbeaten on 76 off 42 balls to take Hyderabad past 200 for the fifth time this season.
