The 30-year-old is to undergo a scan on his injured shoulder to determine the length of his absence, but the EMC Autumn Tests with New Zealand, South Africa and Tonga will likely come too soon.
He will be replaced by Henry Pyrgos as one of four changes to the starting XV which began the win over the Ospreys last Friday. Pyrgos came off the bench at the Liberty Stadium when Cusiter suffered his injury.
Glasgow head coach Gregor Townsend told www.glasgowwarriors.org: "We're clearly disappointed to lose a scrum-half of the calibre of Chris Cusiter for what looks likely to be a significant period of time, but Henry Pyrgos is a player who works very hard on his game, and we'll also be welcoming (Fiji international) Niko Matawalu to the fold in the next few days."
Sean Lamont replaces Tommy Seymour on the wing in the second change among the backs, while Moray Low has recovered from a calf complaint to start at tighthead prop.
Ryan Wilson, who missed the Ospreys game to be present at the birth of his daughter Ava, returns at number eight, with flanker Chris Fusaro rested for the fixture. John Barclay switches to the flank.
Lock Tim Swinson, who signed from Newcastle Falcons in the summer, could make his competitive debut off the bench after recovering from a facial injury sustained in pre-season.
The Warriors lost their opening fixture at their new Scotstoun home when the Scarlets claimed an 18-13 win a fortnight ago. Glasgow have won just one of the last four meetings with Connacht, the 24-3 victory at Firhill in May.
Townsend added: "Connacht are a battle-hardened, very proud unit whose defeat to tournament leaders Scarlets last weekend didn't reflect how well they performed in the game.
"Our recent record against them underlines just how disciplined, aggressive and accurate a performance will be required to follow up our excellent win over Ospreys with another victory this weekend. As a group, we are really energised by the opportunity to give our supporters a first win at Scotstoun, but are fully aware of what it will take to bring about a positive result."
- Chris Cusiter
- Gregor Townsend