Chasing 159, Chennai completed their eighth win to keep hopes alive in the tournament.Now, both teams are in the last four with 17 points each.
Australian batsman Michael Hussey top scored with 56. He hit four sixes and as many fours.
He made 97 runs for the first wicket with Murali Vijay (36). Later, captain Mahindra Singh Dhoni made 28 off 21 balls.
For Kolkata, off-spinner Sunil Narine of the West Indies and pace-man Rajat Bhatia claimed two wickets each.
Earlier, sent into bat by Chennai, Kolkata made 158 for six in the allotted 20 overs after captain Gautam Gambhir (62) and Brendan McCullum (37) provided a solid start of 99 runs.
Slow left-arm bowler Shadab Jakati took two for 26
