Moore, who captured the medalist honors with a 1-under-par total of 139 in two rounds of stroke-play, posted a 4-and-2 victory over Clint Deibert.
Kuehne, who lost to Tiger Woods in the final of this championship in 1994, notched a 4-and-3 triumph over Ryan Keeney.
Moore, the 21-year-old NCAA individual champion from UNLV, is trying for his second United States Golf Association title of the year. He also captured the U.S. Amateur Public Links crown for a second time in July.
Next up for Moore, a Puyallup, Washington native, will be Greg Peterson.
Kuehne, from Dallas, Texas, will face Luke List in round two.
Second-seeded Nick Cook had no problems on Wednesday, advancing with a 5-and-4 rout of Roger Hoit. However, third-seeded Oscar Alvarez of Colombia was eliminated by Canada's James Allenby, who pulled off a 4-and-2 triumph.
The second and third rounds of match play will be contested on Thursday, with the winners advancing to the quarterfinals on Friday. The semifinals are set for Saturday and the 36-hole championship match is scheduled for Sunday.