"Alex has shown tremendous progression in his short time in our league and gained a lot of experience on the playing field last season," Mack said in a statement announcing the deal.
"He is very well liked by his teammates and possesses great leadership qualities both on the field and in the locker-room."
Starting quarterback Buck Pierce has been hit with injuries in the past and was helped off the field after big hits more than once last season, pressing Brink into action.
The six-foot-three 216-pounder played in eight games, including two starts. The 26-year-old completed 89-of-140 passes for 1,023 yards and five touchdowns against four interceptions.
Brink also served as the team's quarterback during short-yardage situations, rushing for 148 yards and six touchdowns.
A former seventh-round draft pick of the NFL's Houston Texans in 2008, Brink originally signed with the Blue Bombers in 2010 and made one start during his rookie season.
With Brink's signing, the Bombers will have all three quarterbacks they started with last season back in 2012.
Pierce re-signed with the team earlier this month. The Bombers are also bringing back Joey Elliott, who started 2011 as the backup but was sidelined with a torn knee ligament early in the season.
The 25-year-old Purdue graduate has played in four games over two seasons with the Bombers. He started twice in 2010 and has completed 45-of-91 passes for 522 yards and two touchdowns.
Brink knows he will have to compete for his snaps.
"I will come into camp as the backup," Brink said. "That's the expectation."
