Moore, of Pullyap, Washington, was one of 13 players to finish under par after the opening 18 holes. He did his damage at the Pittsburgh Field Club, the easier of two courses being used for the two rounds of stroke play.
The 20-year-old will try to tackle the famed Oakmont Country Club on Tuesday.
Oakmont, the site of seven U.S. Opens, is hosting its fifth Amateur championship. The venerable course will stage all six rounds of the match-play portion of the event.
Only one of the 13 rounds under par was posted at Oakmont, as Steve Conway of Irvine, California carded a 1-under 69.
Moore won last year's U.S. Amateur Public Links title and earned a spot in the 2003 Masters. He nearly missed this Amateur championship, failing to enter by the July 2 deadline, but his selection to the 2003 Walker Cup team gave him a second chance.
Zach Atkinson shot a 3-under 67 at Pittsburgh Field Club, while Ben Bunny, Travis Johnson, Jessie Mudd and Peter Tomasulo notched 2-under 68s.
Trip Kuehne, who finished as the runner-up to Tiger Woods at this event in 1994, carded a 69 at the Field Club.
Following Tuesday's second round of stroke play, the field of 312 will be trimmed to 64 for match play, which begins Wednesday.
The 36-hole final is scheduled for Sunday.