McNeill made 10 birdies in his round of 62 that put him in the last group with Koepka. McNeill is a two-time winner on the PGA Tour.
He has never played in the Masters because both wins were opposite-field events that didn't award full FedEx Cup points.
"I'm going for the record to win the most times without qualifying," McNeill said.
That's no longer the case.
The Frys.com Open is the first event on the 2013-14 PGA Tour schedule, which has gone to a wraparound season for the first time in history. It's the start of the FedEx Cup season, offering full points, meaning the winner goes to Augusta National.
Kokrak started his day by holing out from 129 yards in the first fairway for eagle, and he nearly ended his third round the same way. He settled for a short birdie putt and a 68.
Koepka was at 15-under 198. He has played every major once except the Masters, though this is his first regular PGA Tour start. He was between stops in Scotland (Dunhill Links) and Shanghai (BMW Masters) when he received a sponsor exemption to the Frys.com Open.
Jimmy Walker also had a 62 and was three shots behind with Robert Garrigus (68) and Vijay Singh (65), the 50-year-old Fijian making his first PGA Tour start in nearly two months.
Singh last season failed to finish in the top 125 on the money list for the first time in his career. He missed the cut in six of his last 11 tournaments since filing a lawsuit against the PGA Tour for the way it investigated his admission that he used deer antler spray, which was said to contain a banned substance.
Ryo Ishikawa also holed out from the fairway at No. 10 and had a 67. He was among those five shots behind, still in the mix but needing something low to have any chance the way the leaders are playing.
"I don't think you can play overly aggressive," Koepka said. "You can't play not to lose, either. It's conservatively aggressive."
The average score was 68.8, and Koepka had one of the 24 rounds at 67 or better. Only six players were over par, led by Camilo Villegas with a 77 that dropped him out of contention. It could have been worse for the Colombian. He tried playing a shot from the water in front of the ninth green, and the ball hit off the bank and nearly struck him.