"It was awesome," Stroman said. "More importantly just to get the win."
The Blue Jays took two of three to win just their fourth series in the last 27 trips to Tropicana Field.
Stroman (4-0) scattered three hits, walked two and struck out a career-high nine. The Blue Jays' opening day starter is 8-0 in 10 starts since returning from knee surgery last year.
"A special kid," Toronto manager John Gibbons said. "He's got so many things that he can attack you with. He was on today."
After Edwin Encarnacion hit a tiebreaking RBI grounder, Tulowitzki made it 5-1 on his fifth homer.
"To come through on his birthday, it was nice," Tulowitzki said. "He was great out there. It's always fun to play behind him. Pitches with that intensity."
Tulowitzki had been hitless in his previous 10 at-bats in the series and is hitting .172.
"They made a good move to bring in Barney," Rays manager Kevin Cash said. "He gets a big hit and you have that feeling, know that those guys coming up get paid to drive in runs. They capitalized and got the big hits."
Donaldson's ninth homer with one out in the fourth was the Blue Jays' first hit and put Toronto up 1-0.
Evan Longoria tied it at 1 in the sixth on his fifth homer and second in as many days.
Jake Odorizzi gave up one run and two hits over seven innings for the Rays. Since the beginning of the 2014 season, the right-hander has allowed one earned run or fewer in 21 of 34 starts at home.
"As tough as it's been, to get two out of three against those three pitchers down here you feel pretty good," Gibbons said.
Toronto had 15 hits, including eight homers in the series to become the first major league team to be held to 15 or fewer hits and hit eight or more homers in a three-game series.
Blue Jays: RHP Bo Schultz (left hip surgery) gave up three runs, three hits and walked two while getting two outs in his second rehab appearance Saturday night for Class A Dunedin.
CAUGHT BY CONGER
Rays C Hank Conger stopped a stretch where 48 consecutive runners had successfully stole a base against him when he threw out Ezequiel Carrera at second in the fifth. It was Conger's first caught stealing since last May 29.
Longoria is 6 for 12 with two homers off Stroman. ... Donaldson has three hits — all homers — in 14 at-bats against Odorizzi.
OF Michael Bourn is hitting .250 (7 for 28) with four RBIs in seven games with Dunedin. Released by Atlanta at the end of spring training, Bourn signed a minor league contract with the Blue Jays last month.
Blue Jays: Knuckleballer R.A. Dickey (1-3) will go against Texas RHP A.J. Griffin (3-0) on Monday night in the opener of a four-game home series.