Toronto starter Drew Hutchison was taken out of the game due to elbow soreness after throwing only nine pitches.
The Blue Jays then sent a parade of pitchers to the mound and the win eventually went to Carlos Villanueva (2-0), the third of those hurlers, who worked four innings.
Janssen gave up a one-out single in the ninth before recording his sixth save.
Brett Lawrie doubled in a run in the third and the Blue Jays added two unearned scores in the fourth thanks to a throwing error by second baseman Mike Fontenot.
Vance Worley (3-3) gave up all three runs on five hits in seven innings to take the loss.
Philadelphia left 10 men on base and went 0-for-6 with runners in scoring position.