Chasing a modest 161 for victory, Gambhir and Robin Uthappa (47) stitched together a 106-run partnership to provide a perfect winning platform as Kolkata cantered to the target in 18.2 overs.
Starting on cautious note, the duo gradually gained the momentum and punished the Delhi attack led by Mohammad Shami.
Gambhir picked a sliding leg-side delivery from Laxmi Ratan Shukla for a six at mid-wicket region to make his intent clear in the fifth over of the innings as the southpaw got his second half-century of the season.
Uthappa also showed his aggressive side against Jean-Paul Duminy to come down against the South African off-spinner and hit a huge six in the widish long-on region.
The duo gained confidence with every passing delivery to demoralise the Delhi attack as skipper Kevin Pietersen kept juggling his bowling options.
South African left-arm pacer Wayne Parnell finally managed to deny Uthappa his fifty as the Karnataka batsman was caught behind for 47 off 34 deliveries.
Manish Pandey (23 not out) then played the perfect supporting role to make sure that Kolkata do not repeat the Ahmedabad fiasco, where the team lost six wickets for two runs, eventually gifting the match to Rajasthan Royals.
On a day when nothing worked for Delhi and everything went in Gambhir's favour, the left-hander did survive a drop catch and an open run out.
He was finally caught behind while trying to pull a rising Parnell delivery as Dinesh Karthik gloved a beauty in the 18th over. It was left to Kallis (10 not out) to complete the formalities.
With misfields galore, the Delhi fielders did little to help their tired-looking bowlers.