Happ (8-6) struck out a career-high 12 in a season-best eight innings. He allowed two runs and four hits.
''He was fantastic,'' Blue Jays manager John Gibbons said. ''That's a game you normally think you're going to win.''
Happ's only blemish was Joseph's two-run homer in the fourth.
''It was a 2-0 fastball that it was supposed to be away and I pulled it,'' Happ said. ''Just a bad pitch, and he didn't miss it.''
Joseph's drive matched an Orioles record for catchers. Ramon Hernandez (2006) and Gus Triandos (1959) also homered in four straight games.
Chris Davis was the last Baltimore player to go deep in four straight games, accomplishing the feat from July 11-14, 2013.
The Orioles won for the fourth time in five games and lead the AL East by five games over Toronto and the New York Yankees. It was the AL-best 35th road victory for Baltimore, which has won six consecutive series, extending a season-best streak.
''We knew we had to come in and try to take care of business,'' Joseph said of the series with Toronto. ''Taking two of three is gratifying because they're coming after us hard.''
Gonzalez (6-6) allowed four hits in six innings, including Anthony Gose's leadoff homer in the fifth. The right-hander improved to 2-1 with a 2.78 ERA in his last seven starts.
Blue Jays outfielder Melky Cabrera left in the third, two innings after he was hit on the right elbow by a pitch from Gonzalez.
Nolan Reimold took over for Cabrera, who had started as the designated hitter. X-rays on Cabrera's elbow were negative.
Blue Jays: Third baseman Brett Lawrie will be out until at least September with a left oblique strain, an injury that ended his return to the lineup after just three innings. Toronto placed Lawrie on the 15-day disabled list, retroactive to Wednesday, and recalled left-hander Rob Rasmussen from Triple-A Buffalo. Rasmussen was optioned back to Buffalo after the game to make room for infielder Steve Tolleson, who will come off the paternity list Friday.
Orioles: Right-hander Ubaldo Jimenez (right ankle) will be activated off the DL in time to start Saturday's game against St. Louis, manager Buck Showalter said.
Right-hander R.A. Dickey gets the start as the Blue Jays begin a three-game series against AL Central-leading Detroit. Dickey has won two of his past three outings but allowed five runs in seven innings in a loss at Houston last Saturday.
ON THE MEND
Blue Jays first baseman Adam Lind (right foot) doubled twice in two at bats in his first rehab game Thursday and could rejoin the team for the opener of a three-game series at Seattle next Monday, general manager Alex Anthopoulos said. First baseman Edwin Encarnacion (strained right quadriceps) felt good after testing his injury by running hard Thursday, and could begin a rehab assignment this weekend. Anthopoulos said he hopes to have Encarnacion back within a week.
IN NEED OF A GNOME
After being shown a photo of the garden gnome in the shape of Triple-A Norfolk Tides manager Ron Johnson that will be given away at Norfolk's Aug. 15 game, Showalter asked equipment manager Chris Guth to get him one of the bearded dolls. ''Tell them we'll trade a Chris Davis action figure for it,'' Showalter said, in reference to a recent Orioles giveaway.