Right-hander Dominic Leone was recalled from Triple-A to take Osuna's roster spot for Monday's opener at Baltimore, while infielder Ryan Goins will also start the season with the Blue Jays.
Osuna pitched a scoreless inning in Friday's 1-1 exhibition tie with Pittsburgh. The right-hander, who had 36 saves in 42 chances last season, is eligible to return April 9.
Upton was acquired from San Diego last July 26 and hit .196 with four homers and 16 RBIs in 57 games with the Blue Jays. The Padres are responsible for all but $1 million of his $16.45 million salary.
Toronto selected the contract of catcher Jarrod Saltalamacchia from Triple-A and designated right-hander Mike Bolsinger for assignment.