Cueto led off the fifth with a bunt single. One out later, Frazier lined his team-leading 17th homer of the season.
Frazier went into the game tied with Pittsburgh's Andrew McCutchen and Baltimore's Adam Jones for the most home runs in the month of June. Each had seven.
Frazier hit his second home run off Dickey. He faced the knuckleball pitcher in 2012 as a member of the Mets.
"The knuckleball is pretty hard to hit," Frazier said. "It was crazy. It was a 2-0 count. I didn't think I hit it enough to go out. The last one I hit off him was off the boat."
The boat is a replica riverboat high above the batter's eye in straight away center field over 450 feet from home plate.
Cueto (7-5) gave up three runs -- one earned -- and leads the NL with a 1.86 ERA. He gave up seven hits and struck out eight while winning his third straight start.
Cueto limped around the bases on Frazier's home run with a cramp in his hamstring but was able to work through it for the last three frames.
"When Frazier hit the ball, I went back to tag and I felt it," Cueto said through a translator. "It was just a cramp. I felt better than I did last game. I was able to keep the ball down and stay aggressive."
Lawrie sustained a broken right index finger when he was hit by a pitch in the second. The team didn't immediately announce how long the infielder would be out.
Bautista left because of tightness in his left leg. The All-Star outfielder had a single and a sacrifice bunt before leaving.
Aroldis Chapman struck out the side in the ninth for his 13th save. Chapman was charged with the loss on Friday.
Edwin Encarnacion led off the Toronto eighth with his 24th home run, connecting against his former team.
The Reds used Encarnacion's error at first to take a 1-0 lead in the first. With Frazier on first and two outs, Brandon Phillips was safe when Encarnacion couldn't handle third baseman Juan Francisco's throw on a chopper. Jay Bruce converted the error with an RBI single.
The Blue Jays got help from two errors to take a 2-1 lead in the third. Munenori Kawasaki reached on first baseman Joey Votto's throwing error, Melky Cabrera singled and Toronto loaded the bases on Cueto's misplay of Bautista's bunt. One out later, Colby Rasmus lined a two-run single.
The Reds tied the game in the fourth on Bruce's leadoff double, Ryan Ludwick's single and Brayan Pena's run-scoring double-play grounder.
Encarnacion, who a pair of three-run homers Friday night, homered off Cueto into the left-field seats.
Phillips left in the middle of the fifth with a bruised right heel. "He hit his head with a follow through," Price said. "He wanted to stay in but I thought it best to take him out when I saw him in the field." ... The Reds open their second 10-game road trip of the season with the first of three games against the Cubs.
Copyright by STATS LLC and The Associated Press
CINCINNATI, OH - JUNE 22: R.A. Dickey #43 of the Toronto Blue Jays pitches in the third inning of the game against...
36,089 (85.3% full) - % is based on regular season capacity
86 degrees, partly cloudy
Home Plate - Marty Foster, First Base - Rob Drake, Second Base - Alan Porter, Third Base - Joe West
Johnny Cueto is allowing a .148 batting average against his fastball this year, which is 49 points lower than any other qualified starter in MLB. Brett Lawrie has a .259 batting average against fastballs this year, which is 31 points below the league average among qualified hitters.