McIlroy set the previous record of 62 in his 2010 victory, and Brendon de Jonge matched it last year.
Needing to birdie the final two holes for a 59, the top-ranked McIlroy closed with two pars.
McIlroy started the day three shots behind
In all, he had nine birdies during a 10-hole stretch.
"I feel like it's one of these courses I can get on a roll with," McIlroy said.
The highlight of the day came when McIlroy drove the green on the 345-yard, par-4 14th hole for the fourth time in his career — more than any player in tournament history.
McIlroy is looking to become the first two-time winner of the event. At 18-under 198, he's on pace to shatter the tournament record of 16 under set by
Simpson was second, four strokes back after a 68.
Streb shot 71 and was seven shots behind heading into the final round Sunday.
Jeff Maggert birdied the final two holes in heavy rain for a 4-under 68 and sole possession of the lead after the third round of the Champions Tour's Regions Tradition in Birmingham, Ala.
Maggert moved to 14 under with the strong finish after entering the day tied for the top spot with Kevin Sutherland at Shoal Creek. Sutherland was a stroke back. His birdie attempt on No. 18 missed by a couple of inches to set up a tap-in for a 69.
Sutherland was alone at the top until a bogey on No. 16 brought him even with Maggert, who had a bogey-free round. Both birdied No. 17
Fred Funk was third at 18 under afer a 67. He won the tournament in 2008 and 2010.
English duo James Morrison and David Howell shared a two-shot lead after the third round of the Spanish Open in Terrasa, Spain.
Morrison carded a 4-under 68 while Howell shot 69 to leave both on a total of 7-under 209 entering Sunday's final round at the Real Club de Golf El Prat.
Morrison recovered from a double bogey on the second hole by making seven birdies the rest of the way, with one bogey.