Mickelson atoned for a bogey on the final hole of regulation by employing a scintillating third shot from the bunker while replaying No. 18, putting the ball close to the hole and then draining a short putt for a birdie 4.
Grace had a last chance to force another playoff, but couldn't connect on a 25-foot birdie attempt.
With the victory, the American star gained momentum heading into next week's British Open.
He and Grace each finished 72 holes knotted at 17-under-par 271 after carding matching 3-under 69s at Castle Stuart Golf Links.
Mickelson missed a chance to wrap up the win on 72nd hole but couldn't sink a 3-foot birdie putt attempt.
Third-round leader Henrik Stenson fell out of contention with bogeys on three of his final six holes, ending with a 1-over 73 to finish play knotted with J.B. Hansen (71) for third at 15-under.
Tying for fifth one shot further back were Martin Laird (68), John Parry (72) and Gareth Maybin (71).
Nicolas Colsaerts (69) and Raphael Jacquelin (72) shared eighth at minus-13.
Celebrity Families of 2014 [PHOTOS]