The match began five minutes late after a dog invasion but Gaikwad and Conway soon turned the spotlight to cricket with their blazing stand of 110 in Chennai.
Four-time winners Chennai, led by M.S. Dhoni, reached 217-7, a total their bowlers defended by restricting Lucknow to 205-7 and registering their first win of this season.
England's Moeen Ali took four key wickets and was ably supported by fellow spinner Mitchell Santner, who returned impressive figures of 1-21.
But Gaikwad set up the victory as he smashed his way to his second successive half-century this season to entertain a raucous home crowd, which witnessed its team in action live after a long break due to Covid.
The 16th edition of the money-spinning T20 tournament has returned to its home-and-away format after a few years of pandemic-forced closed-door and limited-venue seasons.
Gaikwad tore into the Lucknow attack with a string of sixes including one off England's Mark Wood after the-left handed Conway hit the express bowler for three boundaries in that over.
Lucknow hit back to remove the openers after leg-spinner Ravi Bishnoi sent back Gaikwad and Wood had his revenge on Conway with Krunal Pandya taking a stunning low and diving catch in the deep.
Lucknow kept chipping away with wickets including Ben Stokes out for eight before Ambati Rayudu, who hit an unbeaten 27, took the total past 200.
Dhoni hit two straight sixes off Wood in the final over to go past 5000 IPL runs before he departed.
In reply, Kyle Mayers smashed 53 off 22 balls to start the chase strongly but Lucknow soon lost their way to Moeen's off-spin.
Moeen got Mayers, skipper KL Rahul (20) and Marcus Stoinis (21) to dent the chase and despite Nicholas Pooran's quickfire 32 Lucknow could not recover.
