Moore found himself in a rut early after Price won each of the first two holes of their match to take a 2-up lead. Moore continued to trail around the turn at fell 3-down after Price won the 11th with a birdie.
'It was pretty ugly on the front,' said Moore. 'Not for him so much but for me.'
Moore won the 12th, but Price answered with a par at the par-3 13th to go 3-up again. The momentum changed at the following hole, however, as Price walked off with a double bogey.
Price stumbled to a bogey at the 16th and Moore squared the match with a bogey at the 17th after Price walked away with another double bogey.
Moore, the 21-year-old NCAA individual champion from UNLV, won the match with a par at the 18th to continue his quest for a second USGA title in 2004.
'I had a lot of opportunities and just didn't take advantage of them until those last few holes,' said Moore, who won the U.S. Amateur Public Links championship for the second time in his career this July. 'I tried to stay as patient as I possibly could.'
Moore will face Jason Hartwick on Friday after Hartwick disposed of Tim Hogarth, 3 and 2, to earn a spot in the quarterfinals.
Jeff Overton knocked off Nicholas Thompson, 3 and 2, and will meet Andy Svoboda, who defeated Tyler Docking, 3 and 2.
'Never dreamed of anything like this,' said Svoboda, a member at Winged Foot. 'Really thrilled to make it on to the quarterfinals and looking forward to playing well tomorrow.'
Dayton Rose had to work overtime on Thursday as he needed 20 holes to beat Nick Cook. Rose will face Chris Nallen, who out dueled Chad Poling, 2-up.
Danny Green knocked off Spencer Levin, 2-up, on Thursday. Luke List outlasted John Holmes, 2 and 1, and will go up against Green in the quarterfinals.
The quarterfinals will be contested on Friday with the semifinals set for Saturday. The 36-hole championship match will take place on Sunday.