The 28-year-old, who has been first choice for the Chiefs this season, has torn a pectoral muscle.
Dave Lewis started in place of Chudley in Sunday's win over Ulster while Stuart Townsend came on for his first game outside the Anglo-Welsh Cup.
"You've just got to get on with it, and we'll get on with it with a team that's got some momentum at the moment," head coach Rob Baxter told BBC Radio Devon.
As well as Townsend and Lewis the Chiefs also have highly-rated teenager Jack Maunder, Fiji international Niko Matawalu and veteran Haydn Thomas on the books at scrum-half.
"That's his (Townsend's) championship rugby debut, how well he did in a pressure situation," added Baxter.
"His kicking was fantastic, his defensive work was absolutely top drawer and you look at that and your mouth waters to the future we could have in the scrum-half department."
Exeter scrum-half Will Chudley is out for two months with a chest injury.