McDowell was overwhelmed by some fantastic approach play by Johnson, who raced into a two-up lead after the first two holes.
The American was three up by the time the players reached the fifth hole, but the Northern Irishman was able to claw a hole back to cut his deficit to two.
However, a par at the seventh was good enough to see Johnson go back to three in front as McDowell laboured.
A spectacular iron into the ninth helped him to a birdie, reducing his deficit back to two.
Neither man won a hole on the back nine, as Johnson matched his opponent all the way to record a much-needed victory for USA.