Dan McFarland's side arrived at Scotstoun with intent, and their dominance of the early exchanges was quickly rewarded with a try in the 6th minute.
On the first venture into the Glasgow 22, James Hume grubbered a smart kick in behind the defence which was collected by a scrambling Fagerson a yard from the Glasgow line, but he was penalised for holding on after great pressure from Ulster.
Winning his 100th cap, Billy Burns nudged the ball into touch to set the stage for a 5 metre lineout, and as was the case many times last season, the set piece ended with a Tom Stewart try.
The previous campaign's top scorer found Harry Sheridan with the throw before he scored his 4th of the season, breaking off the back of the maul and making a trademark snipe for the line.
John Cooney, making his first start of the season, was accurate with the boot and split the posts with his first effort of the night to add the conversion.
Ulster maintained the pace and with just 12 minutes on the clock, scrum-half Cooney extended the lead when he darted over from a close range breakdown to evade 3 defenders and touch down before easily slotting another kick to make it 14-0.
Sean Kennedy scored a similar try in reply for the Scots as he nipped over from close range in the 21st minute with Tom Jordan converting as the momentum swung the other way.
Ulster were forced into a long period defending their own 22 on the back of this try, and they worked hard to keep Glasgow out until the stroke of half time when the forwards mauled over and Turner touched down. Jordan coverted to make it all square at the interval.
HT: Glasgow 14-14 Ulster
Early in the second half, young Ulster flankers Harry Sheridan and Reuben Crothers combined well to get themselves in position in defence to hold up Tuipulotu and prevent a try. Alongside debutant James McNabney at 8, they formed a back row combination with an average age of just 21 years old.
However, the Scottish side did score a third when they worked it wide for Kyle Rowe to cross in the corner, as Ulster's Kieran Treadwell was shown a yellow card.
Double Rugby World Cup-winning Springbok prop Steven Kitshoff then came off the bench shortly after to make his debut in white.
It helped Ulster regain some composure, and they soon won a penalty which was slotted by Cooney from close to the halfway line to reduce the deficit and make it 19-17 to the home side.
Glasgow secured the bonus point when Johnny Matthews ran in from the 22 with an hour on the clock, and Jordan converted to extend the lead. Their advantage was reduced with 64 minutes played when Nathan Doak fired over a penalty to make it a one-score game heading into the crucial final minutes. In the 73rd minute the home side pulled away when Matthews scored again and Jordan added the the kick.
Zac Solomon came off the bench to make his Ulster debut as they pushed for the score that could have brought them the losing bonus point, but Glasgow held firm.
FT Glasgow 33-20 Ulster
Scorers
Glasgow
TRY: Kennedy, Turner, Rowe, Matthews (2)
CON: Jordan (4)
PEN:
Ulster
TRY: Stewart, Cooney
CON: Cooney (2)
PEN: Cooney, Doak
Teams
Glasgow
\[1-15\] Jamie Bhatti; George Turner; Zander Fagerson; Greg Peterson; Scott Cummings; Matt Fagerson; Rory Darge; Jack Dempsey; Sean Kennedy; Tom Jordan; Kyle Rowe; Stafford McDowall (C); Sione Tuipulotu; Sebastian Cancelliere: Josh McKay
Replacements: Johnny Matthews; Oli Kebble; Lucio Sordoni; Richie Gray; Sintu Manjezi; Sione Vailanu; Ben Afshar; Duncan Weir
Ulster<br>
(15-9) Will Addison, Rob Baloucoune, James Hume, Luke Marshall, Jacob Stockdale, Billy Burns, John Cooney;
(1-8) Eric O’Sullivan, Tom Stewart, Tom O’Toole, Kieran Treadwell, Iain Henderson (C), Harry Sheridan, Reuben Crothers, James McNabney.
Replacements: Zac Solomon, Steven Kitshoff, Marty Moore, Alan O’Connor, Matty Rea, Dave Shanahan, Nathan Doak, Ben Moxham.