The Englishman made a fantastic start, capitalising on some sloppy play by Mickelson on the opening two holes to establish an early lead.
He missed a putt at the fourth that allowed Mickelson to halve the arrears, and the American was back on level terms at the fifth courtesy of a superb bunker shot.
Rose was undeterred, however, and played a superb shot into the seventh hole to set up an eagle opportunity, which Mickelson conceded.
Mickelson once again got back to all square before both men stiffed their iron shots to the ninth flag, leaving themselves with great opportunities for birdie.
It was Rose who seized the opportunity, sinking for a one-up lead heading onto the back nine.
Birdies at 11 and 14 saw the balance of power shift in favour of Mickelson, as he was now ahead for the first time in the match.
However, Rose's putter caught fire at the 16th, where he holed for a half, before draining birdie putts at 17 and 18 to snatch the point.