Chay Mullins marked his debut by scoring Ulster’s bonus-point try just six minutes after the restart after more good work from McIlroy, and Murphy’s conversion put the visitors 12 points clear.

But indiscipline cost Ulster as Ben Carson, who was drafted into the team to replace Stockdale, was sent to the sin bin for a high tackle on McDowall and, in the following play, McIlroy was also carded for a deliberate knock on to leave Richie Murphy’s side with 13 players.

Glasgow immediately took advantage as Jamie Dobie raced over in the corner after the hosts worked some space, and Horne landed his third conversion of the game.

The hosts were held up over the line from a rolling maul, but they made the most of their momentum when Sam Talakai burrowed over the line to level the game, and Horne nudged Glasgow back in front.

Despite having Carson back on the field, Rowe then raced clear to extend Glasgow’s lead after some soft Ulster defence.

But Rory Darge and Talakai were sent to the sin bin inside the final 10 minutes to leave the hosts with 13 after repeated infringements as Ulster camped on the Glasgow line, and Augustus crashed over before Doak levelled the game.

In late drama, Carson was penalised for holding on at a ruck to give Glasgow a kickable penalty but Horne could only hit the upright.

The final minutes were chaotic as both sides threw away possession in search of a decisive score, but Zac Ward was bundled into touch with the clock in the red to ensure the game ended level.

Glasgow Warriors host Connacht on Friday in their third match of the season while Ulster face Munster in Belfast on Saturday.

Glasgow: McKay; Rowe; McDowall, Tuipulotu; Dobie; Lancaster, Horne; Schickering, Hiddleston, Richardson; Brown, Williamson;Miller, Fraser, Fagerson.

Replacements: Stephen, McBeth, Talakai, Cummings, Darge, Duncan, Reidy, Kuenzle.

Ulster: McIlroy, Mullins, Carson, Benson, Z Ward; Murphy, Devine; O’Sullivan, Stewart, Knox; Izuchukwu, Snyman; McKillop, McCann, B Ward.

Replacements: Walker, McAllister, Bello, Donnell, McNabney, Doak, Flannery, Augustus.