Norris, who turned 34 on Saturday, had 47 saves over the past two seasons for the Los Angeles Angels and St. Louis Cardinals. He went 3-6 with a 3.59 ERA and 28 saves in 64 appearances with St. Louis last year.
Norris won 15 games for Baltimore in 2014, and it was only in the past two seasons that he became almost exclusively a reliever. In 10 big league seasons, Norris is 67-90 with a 4.45 ERA.
Toronto announced the move Wednesday.