BROOKLYN, NY -Chasing a modest 90 for victory at Lord’s, the visitors were taken to victory by Tom Lace’s breezy unbeaten 31 while captain Chris Dent struck 25.
Brathwaite made 21 off 61 deliveries in 1-¾ hours at the crease, posting 34 for the first wicket with Dent and a further 37 for the third wicket Lace.
The West Indies Test captain perished with three runs needed for victory, lbw to seamer James Harris.
Brathwaite got another good start in the first innings when he made 33 but failed to capitalize then.
Resuming on 26 for three, Middlesex was dismissed in their second innings for a paltry 152, with Nick Gubbins top-scoring with 52 and wicketkeeper John Simpson weighing in with 40.
The pair added 63 runs for the sixth wicket before the last five wickets tumbled for 43 runs.
Seamer David Payne followed up his first-innings five-wicket haul with a spell of six for 56 to end with an 11-wicket match haul.
Gloucestershire is unbeaten for the season and top Group 2 with 95 points from four wins and a draw.
Persistent rain at Edgbaston prevented play between Warwickshire and Alzarri Joseph’s Worcestershire, with Miguel Cummins suffering an identical fate at Headingley, where adverse weather forced an entire washout between his Kent and hosts Yorkshire.