Crisp's solo shot in the third inning put the A's ahead 3-2 and Oakland added another run on Barton's RBI single in the fourth for a two-run lead.
The Tigers, however, came back to tie it 4-4 in the fifth on Miguel Cabrera's 43rd homer of the year, a two-run shot that gave him a major league-best 130 RBI.
Oakland grabbed the lead for good in the next inning when Nate Freiman, Alberto Callaspo and Barton got consecutive singles off Jose Alvarez (1-4) and Crisp (3-for-6) followed one out later with another run-scoring hit.
A.J. Griffin (11-9) earned the win despite yielding four runs on seven hits over five full innings, and Grant Balfour nailed down the victory with his 33rd save despite allowing a run in the ninth.
The second-place A's came into the contest trailing American League West-leading Texas by 2 1/2 games. They forged a two-game lead over Cleveland in the AL wild card race with the win.