Despite rallying back with two closing birdies, a bogey at the 18th sealed 30-year-old Harman's fate.
It also marks the seventh consecutive major that has crowned a first time victor.
Of the five other players who shot a 63 in the third round at a major, none went on to win. It's the first time that has happened since 1998-2000. It was close for a while, as Brian Harman kept catching Koepka for the co-lead most of the day. The 27-year-old has always been considered one of the game's most promising young players.
Thomas also is the 29th player to shoot 63 in a major.
Brooks Koepka is in position to win his first major title.
Hideki Matsuyama is second at 12 under. He took that 5-wood you made for me in the basement and hit it 250 dead on a line - a frozen rope to 5 feet from the pin on a par 5.
When Brian Harman's dad dropped him off at football practice when he was a kid, the elder Harman told him not to be disappointed if he didn't get to play a lot.
The week ended with 31 players under par, breaking the U.S. Open record of 28 players at Medinah in 1990.
Scottie Scheffler is the low amateur for this year's U.S. Open.
Six players start the final round within three shots of the lead, all eyeing their first major.
Rickie Fowler putts on the 12th hole on Saturday.
Koepka is playing with Tommy Fleetwood, one group ahead of Harman and Justin Thomas. His best finish at the U.S. Open was third in 2001. China's Li Haotong was having the most trouble, standing 10-over-par for the day after playing 10 holes.
This is his third appearance at the US Open, and the first time he has made the cut.
Thomas was tied for 24th at 2 under par, 5 off the pace after the second round.
"A 63 for a par 72 is a heck of a score", Miller said, "even if it was the Milwaukee Open". He then calmly sank the putt for the record round and to take a two-shot lead into the clubhouse. Harman missed the cut in his previous two U.S. Opens. A week later, at the Memorial, Spieth shot a 66 in the first round before fizzling and tying for 13th. He had birdies at 1, 4, 9, 11, 14, and 15.
Three pars later, it was all over. Fowler's 10 under was only good for a fifth-place tie.
Not that those competing at Erin Hills on Sunday need extra incentive to grab the trophy.
However with strong winds ripping across the exposed, tree-less par-72 layout on Sunday, the odds on another low-scoring day lengthened.
Stricker shot a 69 to move to 2 under going into the final round.
Justin wasn't the only one who tore Erin Hills a new one; pretty much everyone did. He tees off later on Sunday.
Anyway, Thomas starts only one shot behind, but history is not on his side. He two-putted for birdie, and he rolled in a 25-foot birdie putt on the 17th. He tees off later Sunday.
Whoever emerged victorious would definitely have earned it given the conditions which were reminiscent of the final round of the 1992 US Open at Pebble Beach.
Morgan shot 77 and, remarkably, still had the lead at 4-under 212. Even though it will be hard, I don't anticipate this being a blood-letting.
Wind was supposed to be one of the main defenses at Erin Hills, which is the longest U.S. Open course in history but also has the widest fairways the tournament has seen.
But birdies will be harder to come by on Sunday with winds gusting from 25-30 miles per hour (40-48 kph), bending flag sticks and moving balls over hardening greens.
Since shooting a sublime 65 on day one to lead the tournament, Fowler's progress has been less spectacular, but he remains firmly in contention at 10 under par.
The course had been softened by rain earlier in the week, and it finally played the way the USGA wanted in the final round.