Chasing 158, Royals were restricted to 145 for nine in 20 overs. CSK improved to 16 points from 12 matches, keeping Royals in the third spot with 14 points after 13 games.
Royals began their chase process sedately with openers Ajinkya Rahane (23) and Shane Watson (28).
They mixed caution with aggression to build a strong platform for their team. Just as it seemed they had settled into a nice rhythm, Rahane fell to Mohit Sharma in the sixth over.
Steve Smith (4) also didn't last long and Royals slumped to 48 for two in the eighth over. But they received a bigger jolt when Watson fell, caught brilliantly by an acrobatic Dwayne Bravo at long-on off the bowling of Ravindra Jadeja in the 10th over.
Rajasthan's situation worsened 11 balls later. Jadeja struck again to dismiss Karun Nair (10) to reduce Royals to 75 for four.
The asking rate steadily went up and Royals needed 73 runs off the last seven overs. But they just couldn't muster the big hits needed to chase down the total. Jadeja ended with a stunning 4/11 spell.
Earlier, Brendon McCullum's 81 off 61 balls lifted Chennai Super Kings (CSK) to a fighting total of 157/5.