After being forced off following a tackle late in the first quarter, Cross emerged from the race after quarter-time with his shoulder bandaged.
The club confirmed on Monday his season was over.
Football operations manager James Fantasia said he was confident Cross would rehabilitate the shoulder meticulously.
"Daniel is a quality person and the consummate professional when it comes to his body and the way he prepares," Fantasia said.
"We expect that from him again with his rehabilitation this time around."
After Sunday's match, coach Brendan McCartney said the club would miss Cross' presence.
"He’s a brave soldier and we're going to miss him — a fantastic Western Bulldogs person," McCartney said.
"His first reaction to me was he felt bad that he let us down.
"It says pretty much what sort of person Daniel Cross is."
