Rose made the perfect start, shooting birdies on the first two holes to seize the early control. The American nearly struck back at three, but his 10-foot putt narrowly missed the hole.
However, Rose's missed putt at the next hole saw him concede the hole to Mickelson, who now trailed by just one.
Mickelson was level by the time they reached the sixth tee, as his up and down from the bunker and the par-four fourth hole left him with just three feet left for a birdie.
Rose played a superb shot into the seventh hole to give himself a four foot for his eagle, while Mickelson overshot his approach and was off the green.
Mickelson conceded the hole and Rose went back to one up, but the American was soon back on level terms with a birdie at the next hole.
It was proving to be a topsy-turvy encounter, as both men stiffed their iron shots to the ninth flag, leaving themselves with great opportunities for birdie.
Rose's shot was better, and his putt was conceded. Mickelson could not follow him into the hole and the Englishman leads one-up heading onto the back nine.