Hunter Mahan holed out for an eagle on the 18th at Montreux Golf and Country Club to complete a 68 and join Steve Allan in a tie for third at 12-under-par 204. Bill Haas, whose father Jay was recently chosen as a captain's pick for the U.S. Ryder Cup team, was two shots further back at 10-under-par 206.
After the second round was completed early on Saturday, Thatcher began the third round trailing Corey Pavin but jumped out of the gate with birdies on each of his first three holes.
Thatcher, who held the opening-round lead, added a birdie at the par-4 fifth but bogeyed the eighth to fall behind Taylor, who had a quick start of his own.
Taylor birdied the third and followed that up with an eagle at the very next hole. He then birdied the sixth and tallied a birdie at the eighth to move to 15 under, two shots clear of Thatcher.
Taylor, who had yet to drop a shot this week, found trouble at the par-4 15th after a bad drive. He hit his second shot out to the rough and chipped his third shot past the hole.
The 28-year-old struggled on the green and three-putted for a double bogey to fall back to the pack.
Thatcher had his own problems on the 15th with a bogey, but the 27-year-old took advantage of Taylor's mistake and began a charge starting with a birdie at the 16th.
Thatcher then pitched his third shot to 7 feet for a birdie at the 17th and blasted his tee shot down the fairway at the closing hole. He had a wedge left to the green and landed his second shot within a foot of the hole for a tap-in birdie.
'Today's round was a real test,' said Thatcher, who will be in search of his first career victory on the PGA Tour. 'All day. The course was not playing easy.'
Taylor had a short par putt at the last but was unable to convert to fall one shot off the lead after a round of 69.
Scott McCarron carded a 1-under 71 to join Mark Wilson in a tie for sixth place at 9-under-par 207. Joe Ogilvie and Woody Austin followed at 8-under-par 208.
Pavin managed a 76 in the third round to fall seven shots off the pace at 7-under-par 209. He was joined by Scott Simpson in a tie for 10th.