With a 4-2 lead entering the bottom of the ninth, Johnson allowed a single to the Twins' Jason Kubel and walked Kurt Suzuki before getting the first out. The former Orioles closer then walked Aaron Hicks to load the bases and gave up an RBI single to Eduardo Escobar before being removed from the game.
Right-hander Dan Otero entered for Johnson and gave up a sacrifice fly to Minnesota's Brian Dozier to tie the game, 4-4. Otero pitched the final 2 2/3 innings to pick up the win.
Johnson's ERA rose to 18.90 after his third rough outing in five appearances this season. He was charged with two runs in one-third of an inning Wednesday, throwing 15 of his 27 pitches for strikes.
Athletics catcher Derek Norris hit a three-run homer in the top of the 11th to put Oakland back in the lead before Otero got out of a bases-loaded jam in the bottom of the inning for the win.
Oakland starter Jesse Chavez allowed one run and six hits while striking out nine in seven innings.
Dozier hit a solo homer and ended with two RBIs for the Twins, who went 1-for-6 with runners in scoring position and stranded 11 base runners. Jason Kubel also hit a solo homer.
The Sports Network contributed to this article.