Leg-spinner Mason Crane, called into the England Test squad earlier in the day, took 3-21 as the hosts restricted Glamorgan to 118-6 on a slow pitch.
Captain James Vince (60 not out) saw Hampshire home with six overs to spare.
Meanwhile, Middlesex prevailed in a tight finish at Lord's as Sussex, who needed five runs from the last over for victory, could only manage two.
Middlesex posted 147-6 thanks to Stevie Eskinazi's 57 not out from 44 balls and Paul Stirling's 40 from 31, as England seamer Chris Jordan (2-29) impressed.
Sussex began their chase well thanks to Chris Nash (64), but only he and Stiaan van Zyl (34) reached double figures as they slumped from 125-4 in the 17th over to 140-7 - with eight runs needed from the final nine balls of the innings.
That meant a nervy last over, bowled by seamer Tom Helm - but he began with two dot balls, had Jofra Archer caught off the third, and David Wiese could only manage a two off the fifth ball as Helm held his nerve and Middlesex sneaked home by two runs.
It leaves both South Group sides in mid-table on 10 points from 11 games.
Meanwhile, Hampshire's spin-heavy tactics were clear from the start with Crane, fellow leg-spinner Shahid Afridi and slow left-armer Liam Dawson taking a combined 6-52 from their 12 overs of spin.
Dawson - left out of the Test squad in favour of Crane - took the new ball and made an immediate impact, bowling Aneurin Donald with the third ball of the match.
Glamorgan's top order could not get going and only a stand of 53 in the final 6.3 overs between Andrew Salter (37 not out) and Graham Wagg (15 not out) gave them any respectability.
But despite losing debutant wicketkeeper Calvin Dickinson in the first over, Hampshire always looked in control of an easy run chase.
Vince's unbeaten 60 came off just 41 balls and featured 10 fours, while he was ably assisted by Tom Alsop (28 from 19) and George Bailey (25 not out).
It helped Hampshire move to within a point of Glamorgan at the top with two games left.
Gloucestershire who have a game in hand, and Somerset occupy the other quarter-final berths, two points ahead of Sussex, Middlesex and Surrey.
Who will qualify from the T20 Blast South Group?
Hampshire spun their way to an eight-wicket victory over T20 Blast South Group leaders Glamorgan at Southampton.