Acquired from the Yankees in a deal for Johnson during the offseason, Vazquez (5-3) struck out seven and won for the fifth time in seven starts. He hasn't walked a batter since April 25 and has lasted at least seven innings without allowing more than two runs in seven consecutive starts.
Vazquez, who began his major-league career in Montreal seven years ago, is on his third team in three years.
"I got traded for a future Hall of Famer. It's not like I got traded for a rookie," Vazquez said. "But I'm past that."
The Diamondbacks have a half-game lead over San Diego in the NL West after their ninth win in 12 games. Arizona (27-18) didn't earn its 27th victory last season until June 17.
"For the guys that were here last year, last year is irrelevant," Counsell said. "It's a different group."
Vazquez gave up a one-out double to Carlos Guillen in the ninth and then struck out Rondell White and retired Dmitri Young on a grounder to end his 18th career complete game. It was his second this season -- the other came on May 1 at San Diego.
"All I tried to do was stay calm and stay like I was earlier in the game," Vazquez said.
It was Vazquez's seventh career shutout and first since he was with Montreal, against San Francisco on Aug. 18, 2003.
The game began after a rain delay of 3 hours, 14 minutes. The Tigers sold 23,124 tickets, but about half the fans didn't stick around long enough to see the start.
Those who left missed a pair of outstanding pitching performances.
Jason Johnson (2-4) matched Vazquez for seven innings before Royce Clayton hit a leadoff single in the eighth, advanced on a sacrifice and a wild pitch and then scored on Counsell's single to left past a drawn-in infield.
"He has a great sense of the game," Arizona manager Bob Melvin said.
Johnson gave up five hits and three walks while striking out six in eight innings. He hasn't won since April 18 despite some solid outings.
"Except for one start, he's been great," Tigers manager Alan Trammell said. "He's deserved to win each one."
Through seven innings, both teams had just three hits apiece and didn't have more than one in an inning.
"If you like good pitching, you certainly saw it," Trammell said. "Both pitchers were outstanding. It's a shame someone had to lose.
"We've been struggling [offensively], but with good pitching, it allows us to stay in games," he said.
Arizona closer Brandon Lyon, on the disabled list with an elbow strain, threw Sunday, and Melvin said Lyon felt "pretty good." ... After playing Boston in the Hall of Fame game, an exhibition, on Monday in Cooperstown, N.Y., the Tigers stay on the road to play the Yankees and AL-East leading Baltimore. ... Arizona returns home to face San Diego and the Dodgers. ... Vazquez was 0-2 against Detroit, allowing 14 runs in two games, before Sunday. ...
It was Arizona's second shutout this season. The Tigers were blanked for the second time. ... Arizona went 7-3 on its road trip, tying the most wins it has had on a road trip lasting at least 10 games. ... Detroit ended its homestand 4-5 and fell to 20-22 overall.