Glamorgan 334 for 7 (Ingram 115, Cooke 59*, Crane 3-69) beat Hampshire 332 for 6 (Vince 178, Dawson 74, de Lange 5-49) by three wickets
Captain Vince beat Gordon Greenidge's previous best of 177 in 1975 - as he struck an outrageous 178 - which also marked his best career score.
But Ingram crashed a brutal century before Cooke got the hosts over the line with a matchwinning unbeaten 59 from 27 balls, with the visitors chasing 332 with a three wicket victory.
After losing 30 minutes to rain, with the match dropping to 49 overs a side, Glamorgan won the toss and inserted their hosts on a lovely looking batting track.
Rilee Rossouw, hot on the heels of a swashbuckling 156 against Somerset on Wednesday, blasted early boundaries to the fence.
But after a rapid start, where Rossouw scored 35 in 27 balls, Hampshire faltered from 41 without loss to 54 for 3.
Marchant de Lange had Tom Alsop caught behind, Rossouw was bowled with a beautiful ball which clipped the top of off-stump before George Bailey jabbed to point two balls later.
That saw homegrown stars Vince and Dawson aligned in the middle and they rebuilt the innings in textbook style - slowly building up the foundations as they dropped down to almost three runs an over. Vince flashed into life as they settled and reached fifty in 59-balls, while Dawson passed his 3000th List A run.
Vince continued to crack deliveries to the boundary at will, with 18 fours eventually hit, and moved past a fifth career century - brought up in a speedy 98 balls.
Dawson, after clocking an anchoring 74, was bowled by Michael Hogan, ending a record-breaking fourth wicket stand for Hampshire and started a second mini-collapse.
Vince, after dispatching five maximums, was dropped on the fine-leg boundary but finally fell when he was bowled by de Lange as he tried to give himself room - getting out for 178 from 138 balls.
Sean Ervine was the next to go when he slapped to mid-on to hand de Lange his maiden one-day five-wicket haul - ending with figures of 5 for 59.
Ian Holland and Lewis McManus scampered quick runs at the death to give the hosts a handsome 332.
Glamorgan struggled to get going and lost David Lloyd caught behind to Kyle Abbott in the fifth over. Vince continued his fine match by plucking a stunning one-handed catch over his head to catch Will Bragg, before Andrew Salter was stumped off a wide to hand Mason Crane his first scalp of the match.
It all pointed towards a Hampshire win, but Kiran Carlson and Ingram put on 73 for the fourth wicket before the former was tempted down the tack by Crane to be stumped.
South African big hitter Ingram clattered six sixes on his way to a 70-ball century - which saw him pass 7000 List A runs - and set up a thrilling finale.
The Crane and McManus stumping union combined again to see Jacques Rudolph depart for 26 - with the required run-rate still at 12 an over.
Chris Cooke smashed 21 from the 47th over to set up a nail-biting final three overs with 39 still needed.
Ingram edged thickly to backward point to fall for a thrilling 115 but successive maximums from Cooke made it 22 from the last two overs.
Craig Meschede handed Dawson a simple catch at long on, with nine needed from the final six balls, but Cooke chipped a six off Topley to pull off a remarkable win with a ball to spare.