Earlier, in the finish of a game suspended Saturday night because of rain, Brandon Lowe hit a solo home run in the ninth to lift the Rays to a 3-2 win.
Aaron Loup (3-0) got the win in both games.
In a game shortened and scheduled to go seven innings, Adames came up with one out and hit the first pitch from Wilmer Font (1-2) for an opposite-field homer. An automatic runner started at second base under Major League Baseball's extra-innings rule and Adames connected.
''I wanted to be aggressive because I knew he was going to attack the zone,'' Adames said. ''He gave me the right pitch to hit and I put the barrel on it. I was just happy that the ball went out and we got the win.''
The Rays tied the game with two outs in the seventh when Austin Meadows scored from first after Blue Jays right fielder Teoscar Hernandez misplayed Yandy Diaz's single.
Yoshi Tsutsugo hit a three-run homer and Meadows added a sacrifice fly for Tampa Bay, which has won eight of nine and come back from deficits in eight of their 14 wins.
''Right when you think you count us out, we do all the little things right to come back and give ourselves a chance,'' Rays outfielder Kevin Kiermaier said.
Randal Grichuk, Vladimir Guerrero Jr. and Anthony Alford hit home runs for Toronto. The Blue Jays went 2-3 despite hitting 17 homers in their first week at Sahlen Field, the team's temporary home this season and the park of their Triple-A affiliate.
''We're almost there,'' Toronto manager Charlie Montoyo said. ''To get to the next point, we have to play clean baseball, and we haven't been.''
Loup faced two batters in the suspended game and working the seventh inning in the nightcap. Aaron Slegers pitched a perfect eighth inning for his first save.
Toronto starter Matt Shoemaker worked four innings before being ejected along with manager Charlie Montoyo after a contentious exchange with home plate umpire Vic Carapazza.
In the top of the fourth with two outs and two runners on base and Toronto leading 2-0, Shoemaker delivered an inside sinker to Tsutsugo on an 0-2 count. Shoemaker started off the mound, believing he had ended the inning with a strikeout.
Two pitches later, Tsutsugo hit a 2-2 pitch for a three-run homer. Shoemaker then struck out Adames - his seventh of the game - but was ejected after reaching the Toronto dugout.
''Every little thing adds to the frustration, and I need to control my emotions a little bit better,'' Shoemaker said. ''I know I kept going, but I didn't say one thing, in my mind, that I should have been thrown out.''
Guerrero Jr. led off the bottom of the inning with his third homer to tie it and Alford hit a two-run shot one out later to give Toronto a 5-3 lead.
The suspended game resumed in the fourth inning and went the full nine innings.
Lowe, an All-Star last season, homered in his fourth straight game, connecting for a tiebreaking drive off Jordan Romano (1-1).
Lowe also doubled in the seventh, and became the third player in Rays history with a streak of at least eight games with an extra-base hit, joining two players who had nine-game strings: Evan Longoria in 2009 and Wilson Ramos in 2018. Lowe was held to a single in the nightcap.
Nick Anderson pitched a perfect ninth in the first game to record his third save.
Rays: Starting pitcher Yonny Chirinos came off the IL to worked 2 2/3 innings in the nightcap. He allowed five hits and struck out five.
Blue Jays: SS Bo Bichette was put on the injured list with a strained right knee. He is batting .361. INF Santiago Espinal took Bichette's spot on the roster. ''We lost, at this point, one of the best players in baseball,'' Montoyo said. ''Somebody's going to have to pick up the slack. It's a big loss.''
Rays: After an off day, Tampa Bay will start a three-game series against the Yankees in New York, with LHP Blake Snell (1-0, 2.08) scheduled to start the opener on Tuesday.
Blue Jays: Toronto goes back on the road for a three-game series in Baltimore. LHP Hyun Jin Ryu (1-1, 4.05) is scheduled to face the Orioles on Monday for only the second time in his career. His other appearance against the O's came in his fourth career start, on April 20, 2013.
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