Pravin Tambe took the plaudits for the hosts as he took three wickets for the loss of just 10 runs to halt the Super Kings' pursuit of the 159 total.
The Royals were put into bat by their opponents and notched a decent total thanks to opener Ajinkya Rahane, who posted a 56-ball 70, and Shane Watson.
But it looked as though the Super Kings would fight back through Suresh Raina until Tambe took the vital wicket to halt their momentum as they fell 14 runs short.
"One more game to go. I thought we were brilliant, the way we fought back,” said Royals’ Rahul Dravid.
“We were some 10-15 runs short, but to come back and defend was outstanding.
“We have some special players in this team, the way they perform to beat some of the bigger teams is simply great.”