The 21-year-old, who has played 45 games in his three seasons at the club, fractured his fifth metacarpal when getting his finger caught on another player in a contest.
Sydney Swans Head of Physiotherapy and Medical Servies, Damian Raper, described the injury as "unfortunate".
"He (Rowbottom) was one of those unfortunate players who suffered a contact injury in one of our intra-club games," Raper said.
"He was in a contest where he got his finger caught on one of our other players and sustained a fracture to his fifth metacarpal.
Raper said a timeline regarding Rowbottom's return would be determined next week.
"He had some surgery and will be reviewed next week in regards to a return to play timeframe."
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Raper confirmed that 23-year-old midfielder Oliver Florent was another to suffer an injury in an intra-club game.
"It was really unfortunate for Ollie who was training well, but in match play he got tackled by one of his opponents and got his ankle caught underneath him," Raper said.
"He suffered a lateral ankle sprain and spent a couple of days in a boot just to settle that down. We expect him to get back into training next week."
Newly-appointed co-captain Callum Mills is still on a modified training program having slowly built up his loads from the backend of 2021.
"We have spent a fair bit of time with Callum throughout September and October working on his strength," said Raper.
"He started running towards the backend of last year and slowly build up his training loads, and is on a modified training program at the moment and working on getting him back into match play."