"When you're facing a guy like Strasburg, you may get one pitch all game to do something," Avila said.
Anibal Sanchez (9-7) gave up one run, five hits and two walks over seven innings for Detroit. Jose Veras, a day after being acquired from Houston, was perfect in the eighth in his Tigers debut. Joaquin Benoit, who will keep his job as the team's closer, gave up one hit in the ninth inning.
Sanchez, who has given up two earned runs combined in his last three starts, isn't ready to say he's in a groove.
"I need to keep working on my command," he said. "My first pitch for a strike is not there yet and I need that."
Strasburg (5-9) allowed five runs, six hits and three walks while striking out seven over seven innings. Strasburg is winless in his last four starts and has given up at least four earned runs in three of his last five outings.
"I thought he pitched great," Nationals manager Davey Johnson said. "He just made one mistake and Avila crushed it. Until that inning, the way he was cruising, I thought we might get nine innings out of him."
The AL Central-leading Tigers have won four straight and eight of nine to move 15 games over .500 for the first time this season.
Washington started strong, scoring a run on three hits in the first against Sanchez. Bryce Harper led off with a triple and scored on Ryan Zimmerman's single. Then, the Nationals couldn't get much else going offensively.
"We're not swinging the bats that badly, but we never seem to score for Stephen," Johnson said. "I can't explain that."
The Nationals have scored a combined total of nine runs in Strasburg's last four outings.
Miguel Cabrera faced Strasburg for the first time in the regular season with two outs and a runner on base in the first and he hit a 2-0 pitch sharply up the middle for a single, but was stranded when Victor Martinez hit an inning-ending flyout.
Cabrera later, trying to field a grounder in the fifth inning, appeared to aggravate the left hip injury that had recently knocked him out of the lineup, but he stayed in the game.
"It bothered him a little bit," Tigers manager Jim Leyland said.
Andy Dirks hit a game-tying double in the fourth inning and if a fan, sitting in the front row along the third-base line, didn't field the ball with his glove, the Tigers might've gone ahead. Jhonny Peralta was sent back to third base after scoring, and Avila ended the inning with a flyout.
The catcher made up for the missed opportunity a few innings later, sending Strasburg's 2-1 pitch soaring deep into the right-field seats to give the a four-run lead.
"I pitched well, minus one pitch," Strasburg said. "That was supposed to be a fastball away, but it came back over the plate and he put a good swing on it."
NOTES: Tigers GM Dave Dombrowski said before the game that didn't anticipate making another trade before Wednesday's non-waiver deadline, but did in fact make a deal after the game. Detroit acquired IF Jose Iglesias from Boston for OF Avisail Garcia and RHP Brayan Villarreal, adding a much-needed option if Peralta is suspended as part of baseball's Biogenesis drug investigation. ... Johnson said he doesn't expect the Nationals to make a trade before the non-waiver deadline. ... Johnson said pitching coach Steve McCatty, a Detroit native, was still in a hospital, two days after he was taken there to treat an irregular heartbeat and high blood pressure. He is expected to be released Wednesday. "Just being really careful with him," Johnson said. Detroit RHP Justin Verlander (10-8) is scheduled to start against Washington LHP Gio Gonzalez (7-3) on Wednesday afternoon in the series finale. ... The Nationals, who swept a two-game series against Detroit at home in May, fell to 0-4 at Comerica Park. ... Detroit had a grand slam in its last game with Peralta clearing the bases in a 12-4 win that finished off a sweep of the Philadelphia Phillies.