Cabrera hit his 43rd homer, bolstering his pursuit of another Triple Crown, but Barton had a tiebreaking RBI single in Oakland's two-run sixth inning. Crisp went 3 for 6 with a solo homer and is batting .500 (9 for 18) in his last four games.
The A's scored at least one run against each of Detroit's three relievers after Anibal Sanchez was forced out after throwing 112 pitches in five innings.
"We did a good job of being patient," Crisp said. "I think we got his pitch count pretty good."
A.J. Griffin (11-9) earned his first win since July 29 despite allowing four runs and seven hits in five-plus innings. The right-hander was 0-2 with a 3.47 ERA in his previous four starts.
"I was just trying to battle and keep the ball down good," Griffin said. "It's good to win, a good battle. It felt like a playoff atmosphere."
Grant Balfour allowed an unearned run in the ninth but held on for his 33rd save.
"We just didn't pitch good. I don't know what else to tell you," manager Jim Leyland said. "They were swinging the bats good, and we didn't pitch good. Our pitchers have been terrific, but tonight they got beat. No big deal."
Cabrera went deep in the fifth, sending a 3-1 slider from Griffin over the wall in right for a two-run shot that tied it at 4. He leads the AL with a .359 batting average and 130 RBIs, but he trails Baltimore slugger Chris Davis for the home run lead by three.
"Trying to go off the plate, trying to make him chase something," Griffin said. "It was up. If it would have been down, it would have been a better pitch."
Oakland responded in the sixth, jumping all over Jose Alvarez (1-4). Barton, who was called up from Triple-A Sacramento earlier in the day, had the last of three consecutive singles to open the inning, driving in Nate Freiman. Crisp added an RBI single.
"They got some hits off tough pitches, but Anibal has been great for us, and this was just one of those days," Martinez said. "Obviously, no one wants to give up hits and no one wants to leave runners on base, but we're playing against a big league team and they are playing just as hard as we are."
Sanchez allowed four runs and five hits in five innings. He struck out six and walked three.
Crisp also made a diving catch in right-center to rob pinch hitter Tuiasosopo of extra bases in the eighth. ... Oakland placed RF Josh Reddick on the 15-day disabled list with a sprained right wrist. Reddick sprained the wrist on Sunday. Barton took Reddick's place on the roster. Oakland also designated RHP Pat Neshek for assignment. Right-hander Evan Scribner was recalled from Sacramento to take Neshek's roster spot. ... Detroit SS Jose Iglesias returned after missing two games because of soreness above his left elbow. ... The pitching matchup for the second game of the series on Tuesday night is Tigers ace Justin Verlander (12-9, 3.68 ERA) against LHP Tommy Milone (9-9, 4.30 ERA).