McIlroy – who had missed an opportunity to clinch victory at the DP World Tour Championship after holding a 54-hole lead – recovered from a double-bogey seven on the ninth to sink two quick birdies followed by an eagle at the 14th to help set the early pace.
Berger could also have seen a better round, starting with four birdies and then an eagle on the par-five 11th, only for bogeys at the next and on the 18th to prove costly for the American.
Ancer made his move after heading into the turn at one under, the Mexican rolling in successive birdies and again from the 15th to put himself in contention in the 20-man invitational tournament.
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Collin Morikawa, chasing victory for the world number one ranking, shot a 68 to sit in a tie for seventh.
England’s Tyrrell Hatton signed for a three-under 69 and is tied 11th, while Matt Fitzpatrick and Justin Rose sit in a group which includes Jordan Spieth at one under.
McIlroy is expecting more from himself when he gets back out on the Albany course.
“I left quite a few out there,” he said in an interview broadcast on Sky Sports.
“I only played the par-fives in even par, and there are five of them so I didn’t take advantage of those.
“But overall my game is feeling much better, which I am pleased with.
“We are all striving for consistency and trying to bring the best out of ourselves each and every time we tee it up.”