Pearce, 26, was also fined Aus $125,000 (£65,000), of which Aus $50,000 (£26,000) was suspended for two years.
He admitted he had a "problem with alcohol" and entered rehab after the footage emerged in January.
"Now this has been decided, I can get on with my rehab and work each day on making the right choices," he said.
The sanction, which will mean Pearce can return against the Newcastle Knights in the last week of April, was agreed upon by the Roosters and the NRL.
"This was a complex and unprecedented matter that required careful consideration," Roosters chief executive John Lee said.
"It was paramount we delivered the appropriate level of discipline, but neither could we ignore the player welfare component."
Pearce, the son of rugby league legend and NRL commissioner Wayne Pearce, was stripped of the Roosters captaincy after the incident.
Sydney Roosters have suspended scrum-half Mitchell Pearce for eight National Rugby League games after he was filmed committing a lewd act with a dog.