Their second consecutive win propelled Gujarat Lions to the top of the points table.
Chasing a below-par 164 for win, Gujarat Lions rode on a powerful batting display to overwhelm the target with two overs to spare.
Earlier, Faf du Plessis smashed a quick half century before Gujarat Lions pulled things back with fine death bowling display to restrict the Supergiants to 163 for five.
Finch was the initiator as he used his power to great effect and smashed young leg-spinner Murugan Ashwin (2/31) for four boundaries in the sixth over. He then took on Rajat Bhatia and clobbered the medium-pacer for two sixes in the next over to make his intentions clear.
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Finch raced to his 50 in just 33 balls with the help of seven boundaries ans two sixes but soon after he perished trying to hit another boundary, caught by Ishant Shama at fine-leg off Murugan Ashwin.
But the missed stumping did not cost Supergiants much as McCullum got out soon, caught by Du Plessis of Ishant Sharma as the batsman went for an extravagant shot.
But by that time the damage has already been done and therafter Gujarat Lions captain Suresh Raina (24) and Dwayne Bravo (22 off 10) played sensibly to take the hosts home with 12 balls to spare.