Betts hit a no-doubt home run to left in the ninth inning off Toronto closer Ken Giles to record the cycle.
"He's pretty good", Toronto starter Ryan Borucki said with a laugh, when asked about Betts, who hit a single in the first, a triple in the second, a double in the fourth to go along with his homer.
Hernandez's home run in the third increased the Toronto lead to 4-2 before the Red Sox struck for two in the top of the fifth where slugger J.D. Martinez smoked his major league leading 35th home run. After tossing 15 scoreless innings in his first two starts, Eovaldi was lifted after allowing 10 hits and eight runs (4 earned) in just 2 2/3 innings.
"I don't know if you can [stop Betts from hitting], especially because he's hot", Blue Jays manager John Gibbons said of Betts.
Boston's comeback overshadowed the big league debut of Baltimore centre fielder Cedric Mullins, who had three hits, two RBIs and scored twice. "All you can hope is they get themselves I guess".
Toronto then surged in front 3-2 in the second after Boston third baseman Eduardo Nunez flubbed an easy grounder off the bat of Kevin Pillar for an error that allowed Russell Martin to scamper home from third.
Betts Homers In 9th To Complete Cycle
Rick Porcello (14-5) lasted just four innings while giving up seven earned runs and six hits.
Smoak's hit extended his hitting streak to a season team-high 13 games. "They put some good at-bats against him and that's the second game in a row they've done that". He's surrendered 21 career home runs against Minnesota.
In improving to 2-2 in his rookie season, Borucki served up nine hits over his five innings - but held the Red Sox to a respectable four runs.
The 2016 A.L. Cy Young Award victor was uncharacteristically off, issuing two walks to the first two batters he faced.
Holt started the sixth with a home run to make it 8-6.
The Blue Jays will start right-hander Marco Estrada (5-8, 4.65 ERA) against Rays left-hander Blake Snell (12-5, 2.27) in the series opener at the Rogers Centre.
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