The Tigers were down to their final out in the ninth when Victor Martinez got Max Scherzer off the hook with an RBI single against Dodgers closer Kenley Jansen — who had just struck out two-time AL MVP Miguel Cabrera — to make it 2-2. Ian Kinsler ignited the rally with a leadoff double.
Joba Chamberlain (0-1) struck out the side in the bottom of the ninth for Detroit, but walked pinch-hitter Chone Figgins to open the 10th and was replaced by Phil Coke.
Dee Gordon popped out to first baseman Victor Martinez on an attempted sacrifice bunt, but Crawford lined a 2-0 pitch past left fielder Rajai Davis and Figgins scampered home.
J.P. Howell (1-0) pitched a perfect 10th inning.
Austin Jackson homered for the Tigers in an interleague matchup between division winners from last season.
Gordon led off the Dodgers' first with a homer against Scherzer and Justin Turner had a sacrifice fly in the seventh.
Scherzer allowed two runs and eight hits over seven innings and struck out eight.
The Tigers got a scare in the second inning when 38-year-old Torii Hunter tried to make a sliding catch of a foul ball by Tim Federowicz down the right field line and jammed his left knee as the ball landed in the seats.
The nine-time Gold Glove winner, who homered in each of his previous three games, remained down for several minutes while an athletic trainer looked him over and flexed the knee. Hunter gingerly went back to his position, but left the game when his teammates took the field in the fifth. The Tigers called the injury as a contusion.
Don Kelly, who replaced Hunter in right field, couldn't catch up with a slicing double into the corner by Juan Uribe that sent him careening into the lower fence and advanced Matt Kemp to third. Detroit manager Brad Ausmus pulled his infield in against Turner, who delivered Kemp with a flyball to center.
Detroit is 4-2 in its first seaosn under Ausmus, who ended an 18-year career as a big league catcher in 2010 with the Dodgers.
Dan Haren pitched six innings and allowed three hits, including Jackson's one-out homer in the second.
Alex Avila followed with a single — the Tigers last hit until Ian Kinsler's double in the ninth.
The injuries kept mounting for the Dodgers, who just got Kemp off the disabled list last Friday and are expecting to activate Josh Beckett to start Wednesday night's game.
Catcher A.J. Ellis joined Clayton Kershaw, setup man Brian Wilson and Chad Billingsley on the DL, and right fielder Yasiel Puig sat out his second game because of a swollen left thumb.
Ellis underwent arthroscopic surgery on Tuesday to repair a torn meniscus, and is expected to be sidelined four to six weeks. Federowicz was recalled from Triple-A Albuquerque.
NOTES: Ausmus' first two games as a major league manager count on Joe Torre's career record — not his. Ausmus guided the Dodgers to a 5-3 victory over Colorado the day after they clinched the NL West title in 2009, and the following year they lost 7-5 to Arizona two days before Torre managed his final game in the majors. Torre, who will be inducted into the Hall of Fame on July 27, made a habit of anointing one of his players to manage an insignificant game on the final weekend of the season. ... Kemp, who hit his first two homers of the season in Sunday's win over the Giants, was 0 for 2 against Scherzer and is hitless in 18 career at-bats against him.