Woods, who staged a remarkable rally Wednesday to get past first-round opponent J.B. Holmes, built a 4-up lead through 11 holes and downed Arron Oberholser 3 & 2 at the Dove Mountain Gallery Course.
Mickelson, who is second behind Woods in the world rankings, won the opening two holes and then saw Appleby record nine birdies en route to a 2 & 1 victory. In Friday's third round, Appleby will take on Justin Leonard, a 2 & 1 victor over Lee Westwood.
The remaining third-round match-ups will include Paul Casey vs. K.J. Choi; Jonathan Byrd vs. defending champion Henrik Stenson; Rod Pampling vs. V.J. Singh; Steve Stricker vs. Angel Cabrera; and Colin Montgomerie vs. Stewart Cink.