Trailing 2-0 in the five-game series coming into the match, Crisp got the A's off to the start they needed by singling and scoring in the first.
He then made a spectacular catch at the wall in the second to deny Prince Fielder a home run.
Seth Smith homered in the fifth to tack on an insurance run and Brett Anderson pitched six scoreless innings, ensuring the series will need a game four, again in Oakland, with the Tigers leading 2-1.
The San Francisco Giants also forced a fourth game in their National League Divisional Series with Cincinnati Reds, claiming a 2-1 win in ugly fashion.
The Giants needed a passed ball and a Scott Rolen fielding error to score Buster Posey in the 10th, for what proved to be the winning run.