Hampshire 131 for 3 (Vince 69) beat Middlesex 128 (Hafeez 34, Morris 3-22, Abbott 3-25) by seven wickets
The men from the Ageas Bowl came through the Grace Gates knowing they needed four wins out of four to have a chance of making the knockout stages and their South African pace duo all but clinched the first of these as they bundled out the hosts for 128.
James Vince's 69 made short work of the chase, leaving Middlesex, who have been riding high in the South Group standings still looking for a couple of wins to book their own place in the quarter-finals.
Middlesex were put in after losing the toss and struggled from the outset.
Paul Stirling's poor form in T20 this year continued when he became Abbott's first victim, lbw to the first ball of the fourth over.
Dawid Malan and Stevie Eskinazi briefly threatened to post a score, but once the former had edged Morris through to keeper Lewis McManus the Seaxes lost their way.
Eoin Morgan continued the cameos trend, striking two sixes in his 20, before perishing attempting a third from the bowling of the excellent Liam Dawson, who bowled well in tandem with South African debutant Tabraiz Shamsi.
Middlesex debutant and Pakistan Test star Mohammed Hafeez tried to hold things together with 34, sharing a stand of 46 with wicketkeeper John Simpson. But he was bowled by Wood before Abbott dismissed Simpson and Toby Roland-Jones with successive balls.
Morris picked up two late scalps as Middlesex lost their last five wickets for seven runs in 16 balls.
Having top scored with the bat, Hafeez struck an early blow with the ball when Rilee Rossouw drove him straight to Nathan Sowter at cover.
Hampshire skipper Vince, though, looked in ominous form from the get-go, one sumptuous cover-drive underlining his class.
Sam Northeast tried to follow his example only to blast a Roland-Jones delivery straight up in the air and give Stirling a simple catch.
But Dawson, fresh from his unbeaten half-century against Surrey in the County Championship 24 hours earlier, proved a valuable ally to Vince, who went to 50 from 33 balls with his seventh boundary. The 50 partnership came in just 32 deliveries and although Vince holed out on the cover boundary, Hampshire sprinted home with 31 balls to spare.