Piyush Chawla was the pick of the Kolkata bowlers with figures of 3/19 in 4 overs, including the wicket of Glenn Maxwell who scored just 14.
Morne Morkel bowled with a lot of pace and ended with figures of 2/20.
David Miller (13) couldn't do much damage and fell to Umesh Yadav.
Punjab opener Virender Sehwag (72 in 50 balls) scored his first half-century of the season.
Robin Uthappa scored 46 to lay a solid foundation with fellow KKR opener and skipper Gautam Gambhir.
Awana broke Kolkata's 68-run opening partnership by dismissing Robin Uthappa.
Gautam Gambhir anchored Kolkata's chase for an easy win.