Sussex 283 (Nash 132) and 236 for 8 (Nash 64, Brown 42*, van der Gugten 5-73) beat Glamorgan 252 (Archer 4-91) and 263 (Donald 59, Magoffin 3-50) by two wickets
Although Briggs was out for an aggressive 36, Brown and George Garton guided Sussex home with 20 balls remaining. The win lifted Sussex to fourth place but, although they have a game in hand over Essex, they trail by 43 points.
At one stage they had been reduced to 156 for 7 chasing 233 in 62 overs, with Timm van der Gugten taking 5 for 73, his third five-wicket haul of the season, but the eighth-wicket partnership swung the game.
Glamorgan had resumed after only 29.3 overs had been bowled on the third day, on 149 for 2, but soon lost their overnight batsmen Will Bragg and David Lloyd, who had put on 77 for the third wicket.
Despite an attractive innings of 59 from 64 balls from Aneurin Donald, Glamorgan then lost their way as the last five wickets fell for just 17 runs. Donald was behind off Garton, who then dismissed Mark Wallace with his next ball.
Graham Wagg, who had lost the ball when he stuck Briggs into the River Taff, attempted a shot two overs later, but mistimed and was caught at mid-off. The reckless approach continued, and instead of setting Sussex a challenging target, they were left a chase of less than four an over.
Van der Gugten gave Glamorgan the perfect start when he trapped Ed Joyce leg before with the first ball of the innings, but Chis Nash and Luke Wells were seldom in trouble adding 87 for the second wicket before Wells, was lbw to Owen Morgan's left-arm spin for 33.
At tea, Sussex were comfortably placed on 92 for 2, before Wagg dismissed Nash for 64, following his 132 in the first innings. Sussex then lost 3 for 12, as van der Gugten, who had changed ends, and after dismissing Christian Davis leg before, bowled David Wiese and Luke Wright.
When Briggs joined Brown, after Jofra Archer's dismissal, Sussex required another 77, but Briggs, who has played some useful innings recently, was soon into his stride, striking boundaries on both sides of the wicket. He eventually holed out to mid-on but Brown and Garton finished the job.