Set 312 to win in Monday's day-nighter at Junction Oval, Victoria looked a chance when young guns Ollie Peake and Liam Blackford combined for a big-hitting partnership.
But Peake's dismissal sparked a 5-42 collapse that ended the game.
Victoria were dismissed for 241 off 43.5 overs, after two-time defending Sheffield Shield champions SA started that campaign with an innings win over Victoria at the same venue two days earlier.
Gilkes, who switched from NSW this season, rescued SA with 127 from 115 balls at No.5, with 13 fours and four sixes - his first interstate ton in any format.
Wadia, who originally came to Australia as a teenager, earned a BBL contract last summer with the Adelaide Strikers.
His left-arm finger spin was pivotal in putting the breaks on Peake and Blackford, and eventually the pressure that Wadia built resulted in the decisive breakthrough.
He was the pick of the bowlers in this game with 3-37 from 10 overs.
Peake top-scored for Victoria with 73 from 74 balls, including four fours and three sixes.
Victoria were on the brink at 4-85 when Peake and Blackford combined for a 114-run stand that made the run chase interesting.
But Wadia turned the screws and had Blackford dropped twice in the 34th over. Blackford also survived a sharp run-out chance in the following over, but eventually Wadia's tight spell paid rich dividends.
Peake tried to force the run rate again with a pull shot off Wes Agar and holed out to Jake Fraser-McGurk on the boundary.
That left Victoria 5-199 in the 37th over and still a sneaky chance. But once Glenn Maxwell was Wadia's third victim for only seven for the next wicket, SA were in the box seat.
Blackford made 50 from 69 balls, with a four and a six, before Nathan McAndrew had him caught and bowled for the eighth wicket.
Sam Elliott did not bat after suffering a hand injury while fielding and Victoria's innings ended at 9-241 in 43.5 overs.
After Victoria won the toss, SA were in all-sorts at 3-16. Then Gilkes combined with Henry Hunt (42) for a 104-run stand that steadied their innings.
Jake Lehmann made 33 and when Gilkes was dismissed in the 41st over, SA were well-placed at 6-234.
Captain Ben Manenti then contributed a quickfire 35 from 26 balls and Henry Thornton also swung the bat for 25 off 15 to further fatten the total.
Fergus O'Neill and Austin Anlezark took three wickets apiece as SA were dismissed for 311 in 49.3 overs.
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