The 25-year-old developed swelling in his knee when he started running post-season, after finishing the Demons' year with calf and ankle injuries.
It was found he had damage within the joint and required arthroscopic surgery to fix it.
He will now miss the start of the Demons' pre-season, which will officially begin for players with more than five years' experience on November 18.
"The damage has been repaired and he'll be off legs for six to eight weeks and he'll be back running in approximately two months," football manager Josh Mahoney told the club's website.
Dawes was restricted to just 12 games for the Demons this year in his first season at the club after crossing from Collingwood the previous October.
While ruckman Jake Spencer is expected to be in full training by the end of November after surgery on his AC joint, the lay off for defenders Tom McDonald and James Strauss will be longer.
The backmen underwent shoulder reconstructions at the season's end and will join Dawes in returning to training before the end of the year.
"Both are recovering well and are back running and they'll both be in full training prior to Christmas," Mahoney said.
Twitter: @AFL_JenPhelan
