Laird made 10-footers for par on the 16th and 18th holes and had a 3-under 68, giving him a three-shot lead going into the final round over a new pair of the next generation.
Hideki Matsuyama of Japan roared into contention with a 63. Brooks Koepka finally found some fairways and used his power to shoot a 64. They were tied for second with Zach Johnson, who had a 67.
Laird, a Scot who has lived the last 14 years in Scottsdale, was at 13-under 200.
Among those still in the mix was Francesco Molinari, who made a hole-in-one before more than 15,000 rowdy fans on the par-3 16th hole.
Rory McIlroy extended his lead to four shots with a 6-under 66 in the third round of the Dubai Desert Classic in the United Arab Emirates.
The top-ranked McIlroy, who led by a stroke overnight, made just one birdie on the back nine of Emirates Golf Club's Majlis course after five birdies in his first eight holes.
Denmark's Morten Orum Madsen, who shot a 9-under 63 on Friday, was second after a 66 with six birdies.
England's Lee Westwood (69) was third at 14-under 202, six shots adrift of McIlroy.
With a notable double prize not only within reach, but practically in her grip, teen star Lydia Ko had to settle for half of the spoils — though it represented a significant piece of golf history, nonetheless.
Though the New Zealander blew a late lead and had to settle for a share of second place at the LPGA Tour's season opener, where she finished a shot behind Na Yeon Choi, the 17-year-old Ko became the youngest player of either gender to reach No. 1 in the world ranking.
Ko, who held a four-shot lead on the front nine of the inaugural Coates Golf Classic in Ocala, Fla., double-bogeyed the 17th hole during a wild closing stretch, but still eclipsed Tiger Woods' mark as the youngest to reach the game's apex by nearly four years.
Choi closed with a 4-under 68 to finish at 16 under, a stroke ahead of Ko and Jessica Korda. Ko had a 71, and Korda shot 66.