Jacques Kallis was the key, with the veteran South African all-rounder removing the Delhi openers - David Warner and captain Virender Sehwag - before a tidy knock of 30 in his side's reply to helped Kolkata across the line.
"It has been a fantastic season for us," said Kallis.
"We are focusing on trying to end up first or second and get through to the final. Our wicket at Kolkata does not have this sort of carry. It was good to deliver on this sort of wicket."
Gautam Gambhir, the Kokata captain, hit a swift 36 off 21 balls as he and Brendon McCullum (56 off 44) set the tone for Kolkata's chase of 153.
The stand was the fourth consecutive fifty-century partnership in the last six games with McCullum. In doing so, he got the better of Sehwag, his opening partner for India. "This is just the IPL. I have all the respect for him, He is the one I always admire," said Gambhir after the match.
Delhi were playing their first match without the big-hitting Keven Pietersen, who has returned to England, but it looked like they would be able to cope as Sehwag raced to 23 from just nine balls.
He fell leg before to Kallis, who also removed the dangerous Australian Warner, Pietersen's replacement. Delhi eventually made it to 153 with Irfan Pathan top-scoring on 36.
In reply, the early efforts of McCullum, Gambhir and Kallis left Kolkata needing just 26 from 28 balls by the time the New Zealand wicketkeeper fell to Umesh Yadav's bowling.
Manoy Tiwary (8) and Yusuf Pathan (7) saw Kolkata over the line with eight balls to spare.
Delhi gave away 16 extras in their innings and Sehwag said: "150-odd was not a bad total, but our fielding and bowling was not up to the mark.
"When you are giving boundary balls, it is difficult to set a field."
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