Dickey (7-8) retired the first 13 batters he faced, recording five of his six total strikeouts in that span.
James Loney broke up the no-hit bid with a one-out single in the fifth. Dickey allowed a single to Yunel Escobar in the sixth and a walk to Desmond Jennings in the seventh. No other Tampa Bay player reached base.
He ended up with his sixth career shutout, three of which came last year when he won the National League Cy Young Award while with the New York Mets.
Tampa Bay starter Roberto Hernandez (4-9) also pitched well but was touched for six hits and all three Toronto runs. He struck out five and walked one in eight-plus innings.
Jose Bautista led off the Blue Jays fourth with a double and scored on a two-out base hit by Melky Cabrera. Adam Lind hit a two-out home run in the sixth and Edwin Encarnacion led off the ninth with his 22nd home run, a shot that chased Hernandez from the game.
Dickey and Hernandez combined to complete the game in a brisk 2 hours, 10 minutes.