Punjab had faced elimination last night and chasing 179 to win had looked in serious trouble when they lost Gilchrist and Paul Valthaty early in their reply to leave them reeling at 31 for two.
But a stand of 111 between Dinesh Karthick (69) and Shaun Marsh (42) revitalised the innings and they made it home with seven balls to spare in the end to delight Gilchrist.
"We have a game or two in hand on most teams and might sneak up," the Australian said in his post-match TV interview.
As to the performance, he added: "I thought 200-plus was looking likely [in the Kochi innings] but we pegged them back. Really pleased with the bowlers and a comprehensive batting effort. It is always a happy environment when you are winning. McCullum and Mahela were flying but the bowlers and fielders stuck at it.
"Shaun Marsh is the dominator at the moment in our line-up but Dinesh was excellent."
A stand of 93 for the opening wicket between Mahela Jayawardene, the captain, and Brendon McCullum (32) had set up Kochi for a big score, but Punjab's bowling, led by Bipul Sharma (two for 32), pegged them back in the second 10 overs.
The defeat ends Kochi's hopes of reaching the knockout stages of the competition and Jayawardene said: "We did not bowl well after picking up those early wickets. We haven't been consistent, we had our chance in this tournament.
"We need to pick ourselves up and finish off on a good note."