Kings XI, who were without injured captain George Bailey, won the toss and chose to put the Royals into bat in Ahmedebad.
Karun Nair (25) and Deepak Hooda (19) added more rapid runs in the closing stages as the Royals posted 191-6 from their 20 overs.
They were still way behind the required rate before the middle order of David Miller (54), Wriddhiman Saha (19) Axar Patel (12) and Mitchell Johnson (13) all scored quickly to make it an electric finish.
Punjab needed 14 off the last over to win, but they were only able to force a super over as Patel carved away a four off the last ball, with Punjab also finishing on 191-6.
It would the Kings XI to bat first in the super over and they got off to the worst possible start when Miller was trapped lbw by Chris Morris with the first ball.
However, Marsh, who was making his first IPL appearance of the season in place of the injured Bailey, hit three straight boundaries to help the visitors to 15 runs.
With the first ball of the Royals' super over, Johnson clean-bowled Australian international teammate Watson, which brought another Aussie, Smith, to the crease.
Smith's first ball was a four and also a no-ball from Johnson, giving Rajasthan hope of victory, but they were beaten soon after as James Faulkner missed and was run out by alert Punjab wicket-keeper Saha.
Kings XI win just their second match out of five, while Rajasthan suffer their first defeat in six games.