Crisp couldn't get a grip and the Tigers took advantage with two runs that helped them beat Oakland 5-4 Sunday and take a 2-0 lead in their AL division series.
"It was an unfortunate play for them and a fortunate play for us," Detroit Tigers manager Jim Leyland said. "That's the human element of the game."
The Athletics will face elimination in Game 3 on Tuesday.
"They're not going to cash it in, trust me," Leyland said. "We've got a long way to go."
The A's will host the next three games if they make them necessary in the five-game series.
Baseball is using a 2-3 format in the division series because there wasn't enough time to have an extra travel day with an extra wild-card team in both leagues.
After Crisp's error, Oakland responded with two runs in the eighth to get the lead back on a wild pitch and Josh Reddick's solo homer but blew it again in the home half before losing the possibly pivotal game in the ninth inning.
Reliever Ryan Cook got two outs in the eighth, but threw a wild pitch that allowed Don Kelly to score and make it 4-all.
"Both teams made some mistakes that got the other team in the game," A's manager Bob Melvin said.
Kelly ended the game in the ninth with a sacrifice fly off closer Grant Balfour that scored Omar Infante and put the A's in a tough spot.
Oakland hopes it can draw on its comeback experience from late in the regular season.
The AL West champions became the first team in baseball history to win a division or pennant after being behind by at least five games with fewer than 10 games left, capping the remarkable rally with a three-game sweep against Texas that erased a two-game deficit.
"We're not packing it in. We don't do that, or we wouldn't be here," Oakland second baseman Cliff Pennington said. "We'll keep fighting until we're done, one way or the other. We've been doing this all year."
Crisp has been covering a lot of ground in the outfield this season, his third with the A's, and throughout his career that started in Cleveland a decade ago and continued in Boston and Kansas City.
When Cabrera hit a relatively soft fly to center, the Triple Crown slugger was sure two runners were going to be stranded because he expected Crisp to make the catch, even though he was in deep center at the crack of the bat.
"We're lucky," Cabrera acknowledged.
The A's deserved to win the division, coming back to beat the Rangers, and they will have earned a spot in the AL championship series if they can rally in the division series well enough to win three straight.
Oakland is down, but far from out.
Barring a setback, the A's are expecting right-hander Brett Anderson to start in Game 3 in his first outing since straining his right oblique Sept. 19 at Detroit. The Tigers will counter with righty Anibal Sanchez, hoping he can help them advance to the ALCS for the second straight year and third time since 2006 when they swept Oakland.
"We've been doing this all year," Reddick said. "We've got a great pitcher going, and we just need to get him some run support."