Vince overtook former Sussex star Luke Wright to reach 5,054 runs in 169 matches with a superlative 88 off 56 balls not out as Hampshire totted up 156 for four.
But Jacks smashed eight sixes in an incredible riposte 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.
Chris Cooke’s maiden century in the competition led the onslaught as Glamorgan beat Middlesex by 29 runs at Merchant Taylors’ School.
Cooke smashed 113 not out in 41 balls with seven sixes, moving from 50 to 100 in just 12 balls, as the Welsh visitors piled up 238 for three, with Colin Ingram also unbeaten on 92 from 51 deliveries.
Stephen Eskinazi (59) and Joe Cracknell put on an opening stand of 146 in 12 overs for Middlesex but the chase petered out following their dismissals as they finished on 209 for five.
Rob Yates’ T20 best 71 off 53 balls, plus Sam Hain’s unbeaten 48 off 27 deliveries propelled Birmingham Bears to 202 for six in a 21-run win over Northamptonshire Steelbacks at Wantage Road.
Craig Miles then took four for 29 as the Steelbacks were restricted to 181 for seven in their reply, with Ricardo Vasconcelos’ 65 off 39 balls the only score of consequence.
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