The home side were 12 points behind as the first half neared its end, but Craig Gilroy kickstarted their comeback. They ran in five tries and 33 unanswered points after the break as both coaches fully utilised their benches in the first hit out of the season.
Glasgow created the first scoring chance on nine minutes, when centre Grigg came close, but he broke the deadlock on 14 minutes following good work from the pack.
Ulster hit back with a try six minutes later when Gilroy scored in the corner, but Tagive effort restored Glasgow’s lead.
Glasgow continued to impress and their third try came nine minutes from the break when flanker Smith barged over. But Ulster finished the half strongly and tries from Greg Jones and Gilroy reduced Glasgow’s half-time lead to 19-17.
Angus Kernohan struck for Ulster before Chris Fusaro was yellow carded for killing the ball, and Ulster took advantage with two quick tries from Ethan McIlroy and Sam Carter.
Nick Timoney bludgeoned his way over for Ulster’s seventh, before Graham Curtis scored a superb solo try on his debut to round off an impressive win.