Like hitting a grand slam at Yankee Stadium, for once.
"I sense something different," Toronto manager John Gibbons said.
Tulowitzki also homered as the Blue Jays closed within 2½ games of the AL East-leading Yankees.
With two wins in a row, the Blue Jays became the first team to take a set against the Yankees since early July. On Sunday, Toronto will try to sweep them in a series of at least three games for the first time since May 2003.
"Everybody's doing their part. Everybody's being their piece of the puzzle," Price said.
"The future looks good for this team," he added.
A day after R.A. Dickey and the Blue Jays beat New York 2-1 in a 10-inning matchup between the highest-scoring teams in the majors, Price (11-4) dominated.
Price won for the second straight time since Toronto acquired the 2012 AL Cy Young Award winner from Detroit. The lefty gave up three singles and struck out seven.
"That's what we envisioned" with the trade, Gibbons said.
Price had gotten tagged in his last two starts vs. the Yankees to the tune of a 33.26 ERA. Wearing a different uniform, he looked like a different pitcher against them.
In fact, he had taped a handwritten note above his own locker: "If you don't like it, pitch better." The last two words underlined, three times each.
Price pounded his left hand into his glove three times after his final toss, retiring Brendan Ryan on a popup to Smoak with runners at the corners.
Price combined with two relievers on Toronto's first shutout at Yankee Stadium since 2007.
Smoak broke open a scoreless game in the sixth by connecting against Ivan Nova (4-4) for his first career slam. The Blue Jays had been limited to two singles before they loaded the bases with one out on a pair of walks and a hit.
Smoak was batting just .225 when he launched his drive to right-center. Known as a fine-fielding first baseman, he's had a penchant for getting key hits this season despite his low average.
This was the 285th game Toronto played in the old and new Yankee Stadiums since joining the majors in 1977. The Blue Jays had hit five previous slams against New York, all at home, with the last by Tony Fernandez in 2001, STATS said.
"It's pretty cool, especially with all the players and teams that have come through Toronto," Smoak said.
Gibbons didn't get a firsthand look at Smoak's shot -- he was ejected in the third inning for arguing balls and strikes.
Tulowitzki homered in the seventh, his third in 11 games since Toronto got him in a trade with Colorado.
The Blue Jays improved to 6-2 against New York this year, including 4-1 on the road. Last year, Toronto stopped its 17-game losing streak at Yankee Stadium.
"Before, they were good," Yankees outfielder Carlos Beltran said. "Adding Tulowitzki, one of the best shortstops in the game, and Price, one of the best pitchers in the game, of course was good for them."
The Blue Jays hit a whopping 15 foul balls in the first two innings. Toronto has worn down several starters lately by running up their pitch counts. Having Tulowitzki, Josh Donaldson, Jose Bautista and Edwin Encarnacion at the top of the lineup helps, too.
"It's tough to navigate your way through it," Gibbons said.
Blue Jays: Encarnacion extended his hitting streak to a season-high 11 games with a single in the eighth. The hit came right after he appeared to hurt his hand or wrist on a swing. He was checked by a trainer and stayed in the game.
Yankees: OF Brett Gardner didn't start. He's feeling fine, but hadn't had much career success vs. Price.
Blue Jays: RHP Marco Estrada (9-6, 3.40 ERA) hasn't given up more than two earned runs in seven of his last eight starts.
Yankees: RHP Masahiro Tanaka (8-4, 3.84) is 3-1 lifetime vs. Toronto. He lost to the Blue Jays on opening day.
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