After that, the Angels put together just enough offense and more than enough quality relief pitching to secure another gritty win.
Navarro had two hits after replacing Josh Hamilton in the fourth inning as the Angels earned their majors-leading 36th home victory by beating the majors' best road team.
Yet even Navarro wondered whether the Angels could break through against Smyly, who struck out six straight at one point.
"He was making his pitches, working on the outer part of the plate," said Navarro, who drove in Calhoun for the go-ahead run and chased Smyly with a two-out single. "But on that last pitch, he made his pitch, and I was able to do something with it."
Miguel Cabrera homered and Smyly (6-9) had his career-high in strikeouts for the Tigers, who have lost seven of eight in Anaheim.
After Detroit's pitchers combined for 14 strikeouts in Thursday's series opener, Smyly struck out eight of the Angels' first 10 hitters. The left-hander overwhelmed Los Angeles until the sixth, retiring his first 13 batters and striking out nine different Angels.
"Ten out of the first nine hitters struck out, something like that," Angels manager Mike Scioscia said with a grin. "He had great stuff. For us to match him pitch for pitch, that was something."
Everything changed abruptly with consecutive, one-out doubles by Chris Iannetta and Calhoun, followed by Navarro's go-ahead RBI after the Tigers intentionally walked Albert Pujols to get to Los Angeles' reserve outfielder.
"I felt great pretty much the whole game," Smyly said. "I was executing all of my pitches pretty good, and they were the sharpest they've been in a while, so I felt really confident. ... They have a really good lineup, so I knew I had to stay focused every inning because there wasn't much room for error. For me, it's a good start to build off of."
Tyler Skaggs yielded five hits in 5 2/3 resourceful innings for the Angels, who have won 14 of 19. Los Angeles improved the majors' second-best record to 61-41 and pulled within two games of the AL West-leading Oakland Athletics, who lost at Texas.
Los Angeles' bullpen backed up Skaggs with 3 1/3 innings of one-hit relief, retiring nine straight along the way.
Huston Street pitched the ninth for his second save in three appearances for the Angels, wrapping up an all-around encouraging effort for Scioscia's club.
"Those (relievers) had tough hitters in every inning, and they did a terrific job," Scioscia said.
Skaggs struggled much more than Smyly, needing 43 pitches to get through the first two innings alone, but still struck out five.
"His fastball is his best pitch, and his curveball is really good, too," Hunter said. "That was my first time facing him -- I think everybody's first time facing him -- and he did a great job."
Cabrera connected for his 16th homer leading off the fourth inning, putting his second shot in three days into the stands in right field. Los Angeles didn't get a ball out of the infield until Mike Trout's one-out fly ball to right in the fourth.
The Tigers chased Skaggs with consecutive, two-out singles in the sixth. Detroit then loaded the bases when David Freese botched Martinez's soft grounder to third, but Morin fanned Hunter to end it.
Hamilton left the game with soreness in his left knee and was replaced by Navarro in left field. The former AL MVP grounded out weakly in his only at-bat, and he failed to make a play on J.D. Martinez's long fly to left in the second, allowing it to bounce for a double.
Angels LHP C.J. Wilson threw a bullpen session in his recovery from his sprained right ankle. ... Kevin Jepsen struck out the Detroit side in the seventh for the hard-throwing right-hander's 14th consecutive scoreless relief appearance. ... Detroit's Justin Verlander faces Matt Shoemaker on Saturday night at the Big A. ... Trout was 0 for 4, going hitless for just the eighth time in his last 56 games.