The Englishman reached the turn in 29 and finished his first round with 10 birdies and an eagle for a course record at the Amata Spring Country Club.
"I've never shot a 60 before," Westwood said. "A couple of 61s in Germany. Billy (Foster, Westwood's caddie) said my 62 two weeks ago at the Nedbank (Challenge) was as good a round as he's ever seen but he said today was better."
After going 7 under through six holes, Westwood said he had set his sights on a round of 59.
"It was in my mind to get to 13 under but it wasn't to be," he said. "I gave myself a few chances. Left one short on the hole on 11 and one short on 14. Had a good chance on 16 as well.
"I don't think you can be disappointed with a 60. I have to be a fairly harsh critic of myself to walk away disappointed."
Daly, who shot a 65, had a perfect view of Westwood's round as he was in the group behind.
"That was just one hell of a round," Daly said. "I would have never ever dreamed that I would shoot a 7 under let alone a 12 under. My hat that I'm not wearing is off to him because that was just an awesome round of golf he played."
Just over a month after he stormed off the course during the first round of the Australian Open, the 45-year-old Daly had five birdies and an eagle on the par-5 11th.
"It was solid," he said. "I thought 7 under might be a low round, but what Lee did today was incredible."
Masters champion Charl Schwartzel of South Africa had four birdies and a bogey on his last hole to finish with a 68 and join a group of 11 players at 3 under.