Dustin Johnson used birdies on 13 and 14 to end the match as he defeated double champion Zach Johnson 5&4 in the round of 16 to advance to the last eight, where he will take on Sweden's Alex Noren later on Saturday.
Noren was a 3&1 winner over American Brooks Koepka at Austin Country Club.
Five-times major champion Mickelson advanced with a 4&3 victory over in-form Australian Marc Leishman.
Mickelson, who took three holes in a row at the turn to go 5-up, will next take on fellow American Bill Haas, 1-up winner over compatriot Kevin Na.
Spain's Jon Rahm beat American Charles Howell 6&4, Denmark's Soren Kjeldsen overcame American William McGirt 5&4 and Japan's Hideto Tanihara edged out England's Paul Casey 2&1. (Reporting by Gene Cherry in Salvo, North Carolina, editing by Ed Osmond)