Glover won for the first time on the PGA Tour since capturing the 2009 U.S. Open.
Byrd, seeking his second victory of the year, made a 14-foot birdie putt on the par-4 18th hole to force the playoff and he joined Glover in playing the 18th again as the opening hole of the extra session.
Byrd, however, hit his tee shot into a fairway bunker, missed the green and eventually failed to make a 25-foot par putt. Glover hit his second shot to within 20 feet and two-putted for the winning par.
Glover shot a 3-under-par 69 to tie Byrd at 15-under 273 after 72 holes. Byrd, who had the lead when the day began, closed with a 72.
Rory Sabbatini finished alone in third at 274 after shooting a 65 at the Quail Hollow Country Club. Sabbatini missed a spot in the playoff by less than an inch when his birdie putt on the final hole hung on the lip without falling.
Bill Haas was fourth at 275, Kevin Na wound up fifth at 276 and there was a tie for sixth at 277 among Zach Johnson, Bobby Gates and Pat Perez. Phil Mickelson shot a 69 to close the tournament and was among those tied for ninth at 278.