AFPSimpson, the highest-ranked player in the field, sank several putts from between 12 and 15 feet on the way to a six-under-par 65 in relatively calm conditions at the TPC Scottsdale.
The 26-year-old mixed seven birdies with a lone bogey at the par-four sixth to finish a stroke behind compatriot Ryan Palmer, who snatched the outright lead late in the day before play was suspended in fading light.
Americans Derek Lamely, Kevin Na, Chez Reavie and Harrison Frazar, and Australian Jarrod Lyle, carded 66s while South African Trevor Immelman, the 2008 Masters champion, opened with a 67.
Frost had delayed the scheduled tournament start by an hour and 42 players will have to complete the first round on Friday morning when temperatures are forecast to dip even lower.
Simpson, who ended a career-best season on the US PGA Tour last year with two victories and second place in the money list, was delighted with his performance on the greens.
"The thing that kind of held me in there all day was my putting," he said after totalling only 27 putts. "I made a bunch of putts, so I'm excited about that. The one on 17 was the longest, maybe 20, 25 feet."
Simpson, who teed off at the par-four 10th, rocketed up the leaderboard with four consecutive birdies from the par-four 17th.
"That kind of gave me a good jump start in the first round," he said. "Only blemish was at number six. I really didn't feel like I hit a bad shot, I hit a chip that released a lot more than I thought it would. Other than that, it was solid."
Asked whether it felt strange being the highest-ranked player in the field, Simpson replied: "I'm getting used to it, but I don't think about that stuff that much.
"I'm just a believer in you can climb up as quickly as you can go down, so I don't pay attention to it that much."
Phil Mickelson, winner of the Phoenix Open in 1996 and 2005, needed only 24 putts on his way to a four-birdie 68.
"I feel so good with the putter," the American left-hander said. "It's been a few years since the guys out here have seen me putt like this.
"I'm looking forward to continuing this streak and hopefully getting in contention for the weekend because I feel really confident on the greens."
American world number nine Dustin Johnson and compatriot Keegan Bradley, winner of last year's US PGA Championship, also started out with 68s.
It was not a day to remember, though, for former world number one Vijay Singh, who carded a 76, and 2009 Open champion Stewart Cink, who battled to an 83.
