Entering Saturday second-bottom of the A-League Men and with their finals hopes hanging by a thread, early signs weren't promising for the hosts after Glory weathered an early storm and took the lead through Jordan Elsey.
Late first-half goals from Tongo Doumbia and Botic, though, turned the game on its head and secured the hosts three points that moved them into eighth place on the table, just two points back of sixth-placed Sydney.
Glory fell into United's previous slot of 11th.
They took the lead in the 22nd minute when United keeper Jamie Young's save of a headed effort from Elsey wasn't cleared and the defender pounced to fire home.
James Troisi would try to supply an instant reply but the hosts were otherwise frustrated in their attempts to create clear chances.
That was until Doumbia's powerful header off a Neil Kilkenny corner in the 42nd minute, his second goal in as many games, provided a breakthrough.
A week on after a match-winning brace against Newcastle, 21-year-old Botic then added another when Lachlan Wales put a ball on his chest and he volleyed goalward in the third minute of added time.
Glory outshot United and came agonisingly close to an equaliser just past the hour when Adam Taggart slipped Selim Khelifi in, only for the Tunisian to cannon an effort off the post.
Jack Clisby's follow-up was then blocked.
A head clash subsequently forced Botic from the field, United also lost Troisi to injury in the first half, but they nonetheless eased out an important win.
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