Following Yusuf Pathan's bizarre run-out, RR pinned their hopes on Warne and Tanvir.
The duo dealt in singles and doubles in a bid to fetch the winning runs.
Laxmipathy Balaji's wide delivery in the final over cost CSK two extra runs.
With one run needed off the final ball, Tanvir led them to victory by stealing the winning single.
Watson, Tanvir star as RR clinch title
Rajasthan Royals duly silenced the critics by winning the coveted IPL trophy. Shane Watson was named the Player of Tournament as he finished with 472 runs and 17 wickets. His teammate Sohail Tanvir bagged the Purple Cap after having scalped 22 wickets.