Chasing a daunting 226, Delhi built their innings around a commanding opening stand between KL Rahul and Pathum Nissanka. The pair added 110 runs, with Nissanka striking 62 off 33 balls and Rahul anchoring the effort with 75 from 40 deliveries.
Nitish Rana maintained the momentum with a brisk 33 off 17, before a brief wobble threatened to derail the pursuit.
However, Tristan Stubbs and Ashutosh Sharma held their nerve to seal victory with five balls to spare, completing the franchise's highest successful chase in IPL history.
Earlier, Rajasthan Royals had set a formidable target of 226 after being put in to bat. Riyan Parag led the charge with a fluent 90 off 50 balls, marking a return to form after a lean run earlier in the season.
Dhruv Jurel added 42, but it was Donovan Ferreira's explosive unbeaten 47 from just 14 deliveries that provided the decisive late surge.
Mitchell Starc, featuring in his first match of the 2026 campaign, was the pick of Delhi's bowlers with figures of 3 for 40, although his final over proved costly.
The pre-match spotlight had been on teenage prodigy Vaibhav Sooryavanshi's contest with Starc, but the 15-year-old was dismissed cheaply by Kyle Jamieson.
Despite an entertaining contest, the home crowd left disappointed as Axar Patel's Delhi side kept their playoff hopes alive with a statement win.