Smoak had four RBI while Martin went 3-for-4 with three RBI and a walk. Toronto's 23 hits were a season-best. The Blue Jays had a franchise-high 25 hits against Texas on Aug. 9, 1999.
Adam Duvall hit a solo home run for Cincinnati, his third homer in two games and fifth in the past five.
Stroman (6-2) allowed two runs in six innings, improving to 5-0 with a 2.96 ERA in his past eight starts. He has not lost since a home defeat to Boston on April 18.
Dominic Leone pitched two innings of relief and J.P Howell finished.
Duvall hit an RBI single off Stroman in the first, but Reds right-hander Lisalverto Bonilla (0-3) didn't hold the lead for long. Martin answered with a two-run shot in the second, his fourth.
Toronto broke it open with a five-run, bat-around third, highlighted by Tulowitzki's grand slam off reliever Robert Stephenson. It was the second home run of the season for Tulowizki, who was activated off the disabled list last Friday.
Smoak hit his team-leading 12th homer off Stephenson in the fourth.
Bonilla (0-3) matched a season-high by allowing six runs in 2 1/3 innings. He walked a season-worst five. and saw his ERA rise from 6.17 to 7.71.
Blue Jays: LHP J.A. Happ (elbow) will come off the DL to start against the Reds on Tuesday. ... LHP Francisco Liriano (shoulder) allowed three runs, two earned, in 4 1/3 innings in his first rehab start at Triple-A Buffalo on Sunday night. Gibbons said Liriano could be activated off the DL to start against the Yankees this weekend. ... OF Chris Coghlan (lower back spasms) was scratched from the starting lineup and replaced by Carrera.
Reds: RHP Asher Wojciechowski (1-0, 2.25) makes his first start of the season Tuesday, and will face the team that drafted him in 2010. Wojciechowski will be the fifth rookie to starting pitcher used by the Reds this season.
Blue Jays: Happ has not pitched for Toronto since April 16. He allowed three runs and seven hits in three innings in a rehab start with Class A Dunedin last Thursday.
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