Peralta, who missed most of August and September while serving a 50-game suspension for violating baseball's drug rules, was taunted with chants of "Steroids!" and "User!" as he looped a single to center to bring home Miguel Cabrera.
Sanchez struck out 12 including a record-tying four in the first inning but also walked a season-high six and was pulled after six innings and 116 pitches. Al Alburquerque, Jose Veras, Drew Smyly and Benoit stretched the no-hitter into the ninth before Nava ended the no-hitter bid.
That won't matter to the Tigers, who took a 1-0 lead in the best-of-seven series. Boston's Clay Buchholz will face major league wins leader Max Scherzer in Game 2 on Sunday.
It was a day for pitching in the playoffs St. Louis beat the Dodgers 1-0 in the NLCS.
"That tells you the quality of pitching in the postseason," Tigers manager Jim Leyland said. "We've got to do a lot better with our opportunities, but in this one we were able to hold on."
Boston, the majors' highest-scoring team during the regular season, was shut out at Fenway Park in the postseason for the first time in 95 years.
The AL's regular-season ERA champion, Sanchez loaded the bases in the sixth on three walks. But he struck out Stephen Drew to end the inning, coming off the mound with a celebratory arm pump and high leg kick.
Alburquerque pitched a perfect seventh, Veras got two outs and Smyly retired David Ortiz on a fly ball to center to end the eighth.
Boston's Drew kept the score close in the bottom of the eighth, racing into shallow center field to make a juggling, over-the-shoulder catch on Prince Fielder's mistimed hit with runners on second and third.
In all, Lester gave up six hits and a walk in 6 1-3 innings.
Two nights after teammate Justin Verlander took a no-hit bid into the seventh to finish off Oakland in the AL Division Series, Sanchez followed in similar fashion. His four strikeouts in the first inning thanks to a third-strike wild pitch to Shane Victorino tied the major league record for a postseason game set by the Cubs' Orval Overall in the 1908 World Series.
It's the first time in baseball postseason history a team has had back-to-back games in which its starting pitcher had a no-hitter through five innings.
The Tigers got the only run they needed in the sixth after Cabrera drew a one-out walk and moved to second when Fielder was hit by a pitch. Victor Martinez hit a sharp grounder to short for the force at second, but he beat the relay throw to first by inches to keep the inning alive.
Peralta then knocked a 2-2 pitch to center to score Cabrera from third.
