Suzuki came off the bench as the second pinch-hitter in the inning and drove in Daric Barton from first base to break a scoreless deadlock. Eric Sogard later singled to bring Suzuki in with the final run and give the Athletics five wins in their last eight games.
Milone (3-1) kept Chicago at bay with three hits and five strikeouts. Grant Balfour pitched a scoreless ninth for his fifth save of the season.
Gavin Floyd (1-3) took the loss despite allowing just one run on two hits and two walks in 7 1/3 innings.
The defeat ended Chicago's four-game winning streak.
Both teams finished with four hits for the game.