Betts led off the game with a single, before adding a triple in the second inning and a double in the fourth inning all off Jays starter Ryan Borucki.
Eovaldi is 2-0 since coming to Boston and has yet to allow a run in 15 innings through two starts. He earned his second consecutive win and the second of his major league career. Smoak would finish the game with three hits and two RBIs.
Red Sox slugger J.D. Martinez hit his major league-leading 35th home run, a solo drive in the fifth.
It took until August, but Major League Baseball has its first cycle, courtesy of Boston Red Sox outfielder Mookie Betts.
Betts continues to put together a stellar MVP-level campaign, raising his OPS to a ridiculous 1.103 in 2018.
Before Thursday's game, the Blue Jays welcomed 24-year-old pitching prospect Thomas Pannone to the 25-man roster, up from Triple-A Buffalo.
After the Orioles broke a 3-3 tie in the top of the seventh on a solo home run by Renato Nunez, the Rays loaded the bases in the bottom of the inning against Cody Carroll (0-1), who walked two in setting up Bauers' two-out hit to right.
Boston battled back in the fifth to tie the game.
All-Star closer Aroldis Chapman walked two in the ninth before getting a fielder's choice from pinch hitter Isiah Kiner-Falefa and striking out leadoff hitter Choo Shin-soo and Rougned Odor.
He's hit his first career walk-off homer, clubbed his 100 career home run, and he's now leading the Majors in batting average. Brandon Workman then replaced Porcello.
He was summoned to the big-league club in late June with the Blue Jays in need of a starter what with Jaime Garcia and Aaron Sanchez both on the disabled list.
Travis would tally a RBI single in the sixth inning to make the score 8-4. One batter later, Grichuk cashed in the first baseman with a two-run homer that ended Porcello's night, the Red Sox right-hander charged with seven runs in four innings of work.