Rahane hit a half century as RR posted 159/8 in 20 overs. The 41-year-old Tambe then produced an outstanding spell of bowling to derail CSK's chase. The former champions fell short by 14 runs on Friday night.
Tambe took three wickets for 10 runs in 4 overs with 15 dot balls. This was also RR's 13th consecutive victory at Sawai Mansingh Stadium in Jaipur.
RR will now face the winner of second semi-final between Mumbai Indians and Trinidad and Tobago. The final is on October 6 (Sunday) in New Delhi.
Here are the pictures from the first semi-final.
Hat-trick of half centuries for Rahane
Ajinkya Rahane plays a shot during his half century knock. He constructed his third successive fifty in CLT20. Today, he made 70.
Time to celebrate
Rahane raises his bat to the crowd after reaching his half century in the first semi-final of CLT20.
Captain Dravid is ecstatic
RR captain Rahul Dravid (centre) is in joyous mood as he joins Pravin Tambe (right) to celebrate a wicket. Also seen is wicketkeeper Dishant Yagnik.
Ashwin almost won it for CSK
R Ashwin plays a shot during his 46-run knock. He took CSK closer but that was not enough. He and Chris Morris had a 73-run partnership for the 8th wicket.
CSK have a wicket
Chennai Super Kings players come together to celebrate a RR wicket.
'The Wall' is breached
RR captain Rahul Dravid is bowled by Chris Morris for just five runs.
Tambe is the star
Pravin Tambe is congratulated by his teammates after he took a wicket.
Hussey is run out
Rajasthan Royals are happy after getting Michael Hussey run out.
Suresh Raina in action
Suresh Raina sweeps during his 29-run knock. He was dismissed by Tambe.