Gloucestershire 320 (Higgins 73, Bracey 65, Brathwaite 60) and 14 for 0 trail Hampshire 470 (Alsop 149, Holland 114, Dawson 65, Vince 52) by 136 runs
Gloucestershire closed on 14 for 0 after Kraigg Brathwaite and Chris Dent survived the last six overs, but trail the home side by 136, going into the final day having missed the chance to make Hampshire bat again by just one run.
The visitors looked to have frustrated the hosts' push for a third successive victory on a largely attritional day thanks to a slow pitch, before the game burst into life after tea when they lost 5 for 37.
An excellent 73 from Ryan Higgins in a 76-run stand for the sixth wicket with George Hankins, had frustrated Hampshire after they had bowled well early in the day.
Having resumed on 114 for 2, Tom Lace and James Bracey withstood some testing bowling from Kyle Abbott and Mohammed Abbas, who finished with 4 for 41, in the first hour of the morning.
However, Lace, having played with great restraint, couldn't resist playing at a rare wide delivery from Abbas and edged behind to Lewis McManus for 20.
Bracey, whose brilliant early season form has put him in the frame for an England call-up, played beautifully for his 65.
However, he will not want to see his dismissal to Abbas again any time soon after his stumps were rattled by a ball he played no shot to, in a rare lapse of judgement in his four-hour stay at the crease.
Ian Cockbain was then bowled by Abbas for 24 after lunch before Higgins and Hankins guided their side safely to tea at 271 for 5 with the draw looking the most likely outcome.
But Crane then took three quick wickets when he trapped Hankins lbw for 31, rattled the stumps of Higgins after he bottom-edged an attempted sweep, before dismissing Dan Worrall. for five.
Dawson, who toiled away for 41 overs, got the reward his efforts deserved when he dismissed Josh Shaw and Matt Taylor as Gloucestershire fell agonisingly short of not having to bat again. .