James Driscoll, last year's runner-up, was bounced by Japan's Taichiro Kiyota 1-up.
Daniel Summerhays, the youngest participant at 17, defeated Billy Harvey 4-and-3 to advance to Friday's quarterfinals against Robert Hamilton.
Both Quinney and Driscoll advanced to the third round after winning their second round matches held earlier Thursday morning.
The 22-year-old Quinney held off fellow Walker Cup player and U.S. Amateur Public Links champion D.J. Trahan 1-up after converting a three-foot par putt on the 18th.
The defending champion had a 3-up lead after 15 holes but bogeys on the 16th and 17th allowed Trahan to close the gap.
'Those last three holes are difficult and you just somehow wish it on the fairway,' said Quinney. 'He was hitting the fairways and putting pressure on me. I made a lot of birdies so I had some confidence going. I just wanted to figure out how to get into the clubhouse.'
Quinney is looking to become only the fourth player since World War II to successfully defend his title, joining Harvie Ward, Jr. (1955-56), Jay Sigel (1982-83) and Tiger Woods, who won three in row from 1994-1996.
Driscoll showed improvement in his second round play, after struggling to hold off Casey Yates in 19 holes in Wednesday's first round.
'I think I just executed better,' said Driscoll. 'I think I was pretty focused but my swing wasn't quite there.'
The quarterfinals will be held Friday with Quinney meeting Brian Nosler, who defeated Michael Webb 1-up in the third round.
Kiyota will face Manuel Merizdale, who bested Nick Watney in 19 holes.
In Friday's other quarterfinal match, Bubba Dickerson will square off against Michael Sims, who needed 22 holes to get past John Carlson.
Results of Thursdays second round of match play:
Upper Bracket Morning Matches
Robert Hamilton def. Greg Earnhardt - 3 and 2.
Daniel Ozley def. Scott Abbott - 20 holes.
Daniel Summerhays def. Brooks Kelly - 2 and 1.
Billy Harvey def. Jerry Courville - 2 and 1.
James Driscoll def. Casey Yates, Fouke, Ariz. (142) - 6 and 5.
Taichiro Kiyota def. Sandy Horacek - 2 and 1
Nick Watney def. Nicholas Thompson - 7 and 6.
Manuel Merizalde def. Nick Cassini - 5 and 4.
Lower Bracket Morning Matches
Brian Nosler def. Robert Godfrey - 2 and 1.
Michael Webb def. Jamie Elson - 3 and 2.
Kris Mikkelsen def. John Engler - 1 up.
Jeff Quinney def. D.J. Trahan - 1 up.
Michael Sims def. Simon Nash - 1 up.
John Carlson def. Ryan Blaum - 1 up
Camilo Villegas def. Andy Doeden - 1 up
Bubba Dickerson def. Chris Nalen - 1 up
Upper Bracket Afternoon Matches
1:30 p.m. ' Robert Hamilton def. Daniel Ozley - 4 and 3.
1:40 p.m. ' Daniel Summerhays def. Billy Harvey - 4 and 3.
1:50 p.m. ' Taichiro Kiyota def. James Driscoll - 1 up.
2:00 p.m. ' Manuel Merizalde def. Nick Watney - 19 holes.
Lower Bracket Afternoon Matches
2:10 p.m. ' Brian Nosler def. Michael Webb - 1 up.
2:20 p.m. ' Jeff Quinney def. Kris Mikkelson - 1 up.
2:30 p.m. ' Michael Sims def. John Carlson - 1 up.
2:40 p.m. ' Bubba Dickerson def. Camilo Villegas - 2 and 1.
9:00 a.m. ' Robert Hamilton (141) vs. Daniel Summerhays (142).
9:15 a.m. ' Taichiro Kiyota (143) vs. Manuel Merizalde (142)
9:30 a.m. ' Brian Nosler (141) vs. Jeff Quinney (142)
9:45 a.m. ' Michael Sims (135) vs. Bubba Dickerson (141)