The Tigers are trying to reach a second straight AL championship series after losing last year's ALCS in six games to the Rangers.
Detroit captured the AL Central in Oakland last year and is hoping for another clinching party as soon as possible.
Anderson did his job to delay it.
He insisted he was healthy and ready to go — and manager Bob Melvin took his pitcher at his word and gave him a shot in his biggest start yet. Anderson had shown plenty when he returned in August following a 14-month absence recovering from elbow-ligament replacement surgery and made six impressive starts.
Not feeling quite 100 percent, he allowed two hits, struck out six and walked two in six innings. He was on a pitch count of 80 and was done at exactly that, though was never told about it beforehand.
"I don't know how you could expect more than we got out of him tonight," Melvin said.
Next, the reliable bullpen took over.
Ryan Cook pitched the seventh, Sean Doolittle struck out the side in order in the eighth and closer Grant Balfour finished the four-hitter for a save. The A's staff pitched the 11th postseason shutout by the franchise, while the Tigers were blanked for the 13th time in the postseason.
The A's had lost five straight while facing elimination in the postseason, one shy of the longest active streak by the Twins.
But this group has defied expectations ever since the first full workout at spring training back in February when the A's lost third baseman Scott Sizemore to a season-ending knee injury. Opening day starter Brandon McCarthy took a line drive to the head Sept. 5 and needed brain surgery. Starter Bartolo Colon was suspended for 50 games in August for a positive testosterone test.
Oakland became the first team in major league history to win the division or pennant after trailing by five or more games with fewer than 10 to go. The A's were five back of the Rangers with nine left, then won their final six, all at home with sweeps of Seattle and Texas.
Smith hit a towering drive to the deepest part of center field in the fifth for yet another timely home run for the A's, whose 112 homers after the All-Star break led the majors.
"That's how you win postseason baseball games, with pitching and defense and timely hitting," Smith said. "We had that. We got two runs and that's all we needed. Anderson was great and our defense was, too."
Sanchez gave up five hits and two runs in 6 1-3 innings, struck out three and walked two.
NOTES: Smith hit his first postseason homer and third lifetime against Sanchez in 15 at-bats. ... At 24 years, 251 days, Anderson became the fifth-youngest pitcher in Oakland history to make his first career postseason start. ... Both Bay Area teams avoided elimination after the NL West champion San Francisco Giants won at Cincinnati earlier in the night. ... Oakland sold out for the eighth time this year and second straight — the regular-season finale vs. Texas drew 36,067 — including 1,000 standing-room only tickets and extra suite sales. It was the biggest crowd at the Coliseum since drawing 43,974 against the Yankees on Sept. 4, 2005, before the upper decks were tarped.