Indianapolis and New England announced the move on Saturday.
The Colts added the quarterback for depth as starter Andrew Luck recovers from shoulder surgery. New England needed help at wide receiver after Julian Edelman's season-ending torn ACL injury this preseason.
Saturday is NFL cut down day, meaning teams must take their 90 man rosters down to 53 players.
Brissett, 23, completed 34-of-55 passes for 400 yards in three games last season for New England. He started twice, posting a 1-1 record. While he didn't throw a touchdown or an interception, he did have 83 rushing yards and a rushing score on 16 carries. He was the third-string option in New England.
Dorsett, 24, had 528 yards and two touchdowns on 33 receptions last season. He appeared in 15 games, starting seven times. Dorsett had 225 yards and a score in 2015. The wide receiver was the No. 29 pick overall in the 2015 NFL Draft.
Brissett was a third round pick in the 2016 NFL Draft.
"I would say, again, the odds are most likely he [Luck] probably won't open up against the Rams," Irsay told reporters, according to the team website.
"But let me be clear about it -- in our minds, it's something that we haven't ruled out. We're going to see where he's at. It would be awesome [if he can play]. We're not talking Willis Reed or something like that. [Luck] is a young guy, 12 to 14 to 16, maybe 18 years, I don't know, going forward [with the Colts]. The longer the better, in my opinion."
It's too early to tell exactly how Dorsett will be used in New England, but he is worth a late-round dart throw for your fantasy football team in your draft. He is likely a low-end WR 3 option this season in deeper leagues.
The Patriots and Colts don't play each other during the regular season.