Burnett appeared in 67 games for the Minnesota Twins last season, posting a 4-4 record with a 3.52 earned-runs average.
He's been in the majors three season -- all with Minnesota -- recording an 8-11 mark and a 4.61 ERA in 174 relief appearances.
Robinson was claimed from the Pittsburgh Pirates, but made his MLB debut last year with Kansas City. The six-foot-five, 238 pound left-handed hitter appeared in four games for the Royals, going 0-4 in four pinch hit appearances, but spent the majority of 2012 with triple-A Omaha, batting .292 with 37 doubles, 13 home runs and 67 RBIs.
The Jays optioned Robinson to double-A New Hampshire of the Eastern League, while Burnett was optioned to triple-A Buffalo.
To make room on the 40-man roster, Toronto designated first baseman Lars Anderson for assignment.