The win lowered Oakland's magic number for locking up the American League West title to four.
With the score tied 6-6, Daric Barton walked off Minnesota reliever Shairon Martis to open the Oakland eighth. Martis struck out Alberto Callaspo but Crisp followed with a high drive into the right-field seats.
It was his 21st home run of the season and 11th in his last 27 games.
It was also the only hit Martis (0-1) allowed over 1 2/3 innings of relief.
Sean Doolittle (5-5), who stopped a Minnesota rally in the eighth, collected the win. He didn't allow a base runner over 1 1/3 innings of relief.
Oakland built a 6-3 lead with a four-run sixth in which Callaspo hit a pinch-hit solo home run and Jed Lowrie connected for a three-run homer. But Minnesota came back with an Oswaldo Arcia home run in the seventh and RBIs by Chris Herrmann and Brian Dozier in the eighth to tie the score.
Minnesota opened the scoring with Chris Parmelee scoring in a fielder's choice in the third. Josh Reddick doubled in Brandon Moss and scored on a Stephen Vogt sacrifice fly to put the A's ahead 2-1 in the fourth. Josmil Pinto and Pedro Florimon had run-scoring singles for the Twins in the sixth that left the score 3-2 Minnesota.