The Hawks started the day sitting in sixth place on the South group table, needing to chase down fourth-placed Surrey's run rate to have any chance of progressing - or hope Sussex or Gloucestershire lose their matches on Sunday night.
But the emphatic fifth straight win moved the 2010 and 2012 champions up to third spot and they will now progress if Sussex lose to Kent or Gloucestershire are defeated by Somerset this evening, writes IAN McCULLOUGH.
After being set an imposing 185 target for victory by Glamorgan, who were put into bat by James Vince the home side needed to usurp that total inside 14.1 overs to leapfrog fourth-place Surrey.
Remarkably, they did it in just 13 overs with D’Arcy Short and Joe Weatherley leading a sensational run chase.
Short struck 69 off 30 balls, reaching his half-century in just 18 deliveries - there have only ever been four quicker fifties scored in the 18-year history of the T20 tournament.
Weatherley, meanwhile, smashed a startling unbeaten 43 from just 13 balls in just 19 minutes.
After being asked to bat, the Welshmen took full advantage of the perfect conditions despite losing opener Kiran Carlson to the third ball of the match with David Lloyd and Australian Test batsman Marnus Labuschagne building an impressive 92-ball partnership.
Lloyd belted four sixes in his 52 off 33 balls before Labuschagne continued the assault against the home attack, clearing the rope six times in an entertaining 78 from 47 deliveries before falling to Brad Wheal, who finished with 3-39.
Dan Douthwaite added some extra gloss to the total, smashing 21 from eight balls to help the visitors finish their final innings of a disappointing campaign with a flourish.
However, Vince and Short showed they meant business from the off, with the Australian hitting three successive sixes off Timm van der Gugten's first over that went for 26 runs with the pair than taking 21 off Roman Walker's first over.
Short brought up his 50 off just 18 balls as the Hawks completed the power play at 79-0 before Vince departed for 39 when he was caught by Jack Cooke off Andrew Salter.
Big-hitting Colin de Grandhomme came in at three to try and continue the run bonanza, but he was caught on the boundary by Carlson off Labuschagne for five shortly before Short was dismissed by the spin of Salter.
Those departures looked to be a hammer blow for the home side's hopes, but Weatherley stepped up to the plate and sealed a memorable victory with four sixes and three boundaries.
After winning just one of their first nine games - there were five defeats and three rain-enforced cancellations - Hampshire ended with five straight successes to raise hopes of a first quarter-final since 2017.