The Warriors return home after opening their campaign with a bonus-point victory over Munster at Thomond Park, with Franco Smith's squad now looking to build on that result in their first competitive outing at Scotstoun this season.
Glasgow won all eight of their URC regular-season matches at Scotstoun last season, and Forrest believes backing up last weekend's result will be key.
"We do know it's a very proud record we've got at home," the Assistant Coach told BBC Scotland.
"We've just gone away to one of the hardest places to go in Munster and we've come away with a win, even though we were a little bit rusty in some areas.
"We've got a group of players that know how to win and how to see out games. The biggest thing for us this weekend is back that up. That win doesn't count for much if you then come home and you cough up a result.
"We've taken five points on the road, that gets made even sweeter if we back that up with another five points this Saturday."
The Warriors will also continue to manage the workload of their international contingent throughout the campaign, with Scotland captain Sione Tuipulotu having been rested for the victory in Limerick.
Tuipulotu is back in contention for selection against Ulster, with Forrest highlighting the demands facing Glasgow's Scotland internationals in a season that leads into next year's Rugby World Cup.
"He played a lot last season," said Forrest.
"This is a big season that's going to lead into a World Cup next October so there's a big piece here where all of our Scotland players are going to get managed at points.
"It's tough, there's no point shying away from that, when you've got 20, potentially 20-plus, players that have to perform here and perform for Scotland in such an important year.
"Even the autumn games, that's completely different now that it's the Nations Championship. All those fixtures mean something, there's a finals weekend off the back of it. You look historically where we maybe played a Portugal or a Tonga in that window and Scotland could rotate, that now changes things.
"Then obviously there's an eye on the World Cup. So yes, it's tough, but when you look back on the way [head coach] Franco Smith has selected teams in the past, we always rotate players. There is a trust in the whole squad."
That approach will again be important as the Warriors balance their domestic and European commitments with the demands placed on their international players throughout the campaign.
Saturday's meeting with Ulster represents the first opportunity of the season for Glasgow to extend a Scotstoun URC record that remained perfect throughout the 2025/26 regular season.