The 19-year-old, the younger brother of Ryder Cup player Matt Fitzpatrick, was two down at the sixth after his American opponent shot back-to-back birdies on the fourth and fifth holes.
Fitzpatrick won the eighth with a birdie to move within one, but Hammer birdied the ninth.
That left the Englishman two down again as they headed back down at the Spyglass Hill Golf Course in Pebble Beach, California.
Hammer, seeded number two and ranked 17th in the world amateur ranking, parred the 12th to move to three up when Fitzpatrick had a bogey.
Fitzpatrick - who as a 14-year-old caddied for Matt when he won the event in 2013 - again narrowed the lead, winning the 15th hole, but Hammer finished with a birdie on the 16th to win 3&2.
Norway’s Viktor Hovland, world number five amateur, progressed to the semi-finals after winning seven holes in a row, beating American Austin Squires 7&6.
Isaiah Salinda also makes up the last four, beating compatriot Will Gordon 2&1, while Devon Bling, competing in his third USGA championship, finished one up to knock out Davis Riley.