Looking for their first win at Scotstoun in more than five years, Ulster made the perfect start when Bryn Ward crashed over in the fourth minute, but Josh McKay hit back for Glasgow.
The hosts were reduced to 14 men when Dan Lancaster was sent to the sin-bin for a deliberate knock-on, and Ulster immediately capitalised when Ethan McIlroy crossed.
Despite being a player down, Glasgow hit back through Kyle Rowe’s try, but Ulster held a five-point lead at the break when Jack Murphy scored in the final play of the half after the hosts had a penalty overturned.
Ulster were sitting pretty when Chay Mullins secured the try bonus point, but centre Ben Carson was sent to the sin-bin for a high tackle and McIlroy followed in the next play for a deliberate knock on.
Glasgow scored three tries in a nine-minute burst through Jamie Dobie, Sam Talakai and Rowe, and looked set to surge clear for back-to-back wins.
But there was late drama as Rory Darge and Talakai were sin-binned inside the final 10 minutes after repeated Glasgow infringements on their own line, and Nathan Doak converted Juarno Augustus’ try to level the game.
George Horne struck the post with a penalty with four minutes left, and the sides had to settle for a draw only after Ulster Wing Zac Ward was tackled into touch to end the game.
