Vince faced 151 deliveries before being dismissed late in the day for 42, before injured tailenders Kyle Abbott and Brad Wheal held out for the final six overs to deny the Tykes a crucial victory on the final day of the LV = County Championship Division One clash.
Hampshire coach Adi Birrell set his players a target of each soaking up 72 balls to save the match on a lifeless day four pitch just as Surrey and Gloucestershire had against the hosts earlier this season.
Nightwatchman Mason Crane and Vince fulfilled their side of the bargain as Hampshire finished the day 177-9 with Wheal and Abbot unbeaten at the close, in pursuit of an unlikely target of 392 for victory.
Crane repelled 196 balls - the most he has faced in a single innings - before falling for 28 and his exit just after tea saw the Tykes take three wickets in eight overs with the new ball to put themselves in view of victory.
But Vince, shelved his natural attacking game and showed magnificent technique and temperament to keep alive his side’s County Championship title aspirations.
Starting the day on 26-2 with Joe Weatherley and Crane at the crease, the pair survived the first 28 overs of the day until Weatherley fell lbw on the stroke of lunch to the impressive Jordan Thompson.
Nick Gubbins exited after the restart for six when he was caught and bowled by Andrew Fisher, but with runs irrelevant, Vince joined the dogged Crane at the crease and the pair made it safely to tea with just 44 runs added from 29 overs.
The docile pitch continued to offer little encouragement for both the spin of Dom Bess or the tireless Yorkshire seamers.
Fisher finally made the breakthrough with the new ball shortly after the restart when he brushed the edge of Crane’s bat with Adam Lyth taking the catch at slip to end his stubborn vigil.
Fisher struck again in his next over to pin Liam Dawson on the crease and trap him lbw for a duck to leave Yorkshire in sight of victory.
Thompson, as he does so often for his side, made something happen when his side needed him most, when he bowled Lewis McManus to leave Hampshire on the brink at 138-7 with 25 overs remaining.
Keith Barker survived nine overs before falling to the excellent Thompson bringing Abbott, who was unable to bowl on day three due to a foot injury, to the crease with,16.3 overs remaining to join Vince who was then dismissed six overs from the close by Thompson.
Abbott was joined by fellow South African Wheal who held out magnificently with 10 men around the bat for the final three overs as Yorkshire fell short in an absorbing finish.