Right-hander Edwin Jackson was also placed on the IL with what the team called a lower back strain. Outfielder Billy McKinney and right-hander Justin Shafer were recalled from Triple-A Buffalo in corresponding moves.
Smoak hasn't played since Friday and missed the end of Toronto's series in Houston with what was initially said to be a hamstring issue. He expressed hope over the weekend that an IL stint could be avoided, Rob Longley of the Toronto Sun reported Sunday.
The move is retroactive to June 15, meaning Smoak will be eligible to return as early as next Tuesday, Sportsnet 590's Mike Wilner reports.
Smoak, 32, was thought to be one of the Blue Jays' top trade pieces ahead of the July 31 deadline and is hitting .225/.364/.423 with 12 homers and a team-high 34 RBIs through 62 games this season.
Jackson, meanwhile, hit the IL less than 24 hours after his latest rough outing. After entering Monday's game in the second inning behind an opener, he was tagged by the Los Angeles Angels for seven earned runs in just two-thirds of an inning. The 35-year-old's ERA as a Blue Jay now sits at 12.43 with a 2.29 WHIP in only 25 1/3 innings.
"I think my pitching has just been horseshit," Jackson said after Monday's game.
McKinney, who began the season on the Blue Jays' roster but struggled at the plate, returns to the big club nearly a month after he was optioned. He put up a .717 OPS with two homers in 21 games during his stint with Buffalo.
Shafer, meanwhile, was recalled barely 48 hours after having been sent down in a separate transaction. He owns a 3.86 ERA in five big-league appearances this year.