Surrey 157 for 4 (Jacks 83*) beat Hampshire 156 for 4 (Vince 88*) by six wickets
Vince overtook former Sussex star Luke Wright to reach 5054 runs in 169 matches with a superlative-ridden 88 not out as Hampshire totted up 156.
But Jacks smashed eight maximums in an incredible riposte in a match dominated by one batter on either side as Surrey extended their unbeaten run, going back to 2015, over Hampshire to 11 matches.
For Surrey it got them back to winning ways after defeat to Sussex Sharks, while defending champions Hampshire have lost two of their first three matches.
Having been stuck in, Vince was forced to wait 17 balls before he finally faced a ball, and he only saw eight balls in the Powerplay - which saw Ben McDermott smash 18 before skying to square leg.
Even when he got a go, things were less than fluent between him and Tom Prest, with a pacy hybrid pitch helping the ball skid past the edges and made timing tricky.
There were hallmark Vince moments in there though, a powerful cover drive some swiveled pulls and he earned a life on 13 when Chris Jordan's full stretch over his shoulder only pushed to the boundary.
His fifty, his 57th in T20s, came in 37 balls, with the 83 stand with scratchy Prest forming the basis of the Hampshire total. Prest would get stumped for a run-a-ball 33 and Joe Weatherley and Ross Whiteley came and went but the moment was Vince's.
He overtook Wright in the most Vince way possible, a cover drive. When the stadium announcer Kris Temple informed the crowd of the record he reticently waved before dispatching the largest of his three evening sixes as a more apt celebration.
He is now the 18th-highest run scorer in T20 history, and a rare name on the top run scorer list never to have played in the IPL. He boasts the two best Blast seasons. And depending on the future of the Blast, his record may not be bettered.
Vince ended up with exactly the same unbeaten 88 as against Middlesex on Friday night, and had almost single-handily guided Hampshire to a slightly under-par score - despite facing less than half the balls.
Laurie Evans and Sam Curran also struggled to time the ball consistently and came and went, but Jacks found ways to power over the boundary. He flicked James Fuller over square leg before giving himself space to pump Nathan Ellis over extra cover.
Mason Crane was next for the Jacks six-hitting showcase with a pair of heaves leg side as he arrived at his half-century in 36 balls, but soon saw Sam Curran stumped.
Sunil Narine hit a six then holed out but Jacks continued his onslaught of Crane with a trio of sixes to swing the game toward the visitors. Jamie Smith assisted the cause with a 64-run stand with Jacks. It was fitting that Jacks scored the winning run with seven balls to spare.