Resuming on their overnight total of 171-6 following a middle-order collapse on the third day, and chasing an improbable 556 for victory, the hosts ultimately suffered a 256-run loss.
Seventh-wicket pair Charlie Hemphrey and Graham Wagg resisted for 50 minutes, sharing a partnership of 62 before the former's five-and-a-quarter-hour vigil ended.
Opener Hemphrey had faced 254 balls, striking a six and eight fours in his 72 before feathering an innocuous Toby Roland-Jones delivery down the legside to wicketkeeper John Simpson.
It was a disappointing end to a previously chanceless inning, but Hemphrey at least showed he has all the attributes to develop into a successful opener as he comes to terms with his first season in county cricket.
Wagg followed just 13 balls later, guiding another Roland-Jones delivery to Dawid Malan at second slip, with a six and seven fours in his 60-ball 40.
Staring defeat in the face, Marchant De Lange decided it was time to strike some lusty blows, notably scoring 24 – two sixes and three fours – off Tim Murtagh's only over of the morning before he was appreciably replaced.
Roland Jones then removed the off stump of Lukas Carey to claim his fifth wicket of the match and finish with match figures of nine for 113.
The former England seamer has been a revelation since returning from injury and indifferent form, claiming 19 wickets overall in his last two games.
Following a rapid-fire partnership of 35 off five overs between De Lange and Michael Hogan, Glamorgan were eventually all out for 299.
With five fours in his 22, Hogan was caught on the long-off boundary by George Scott, off the bowling of Nathan Sowter, leaving De Lange undefeated on 45 off 28 balls, that included five fours and three sixes.