McDowell, the 2010 U.S. Open champion at Pebble Beach, birdied the final three holes despite strong afternoon winds sweeping the course next to the Red Sea.
Five players were one shot behind at 5 under, including 2016 British Open champion Henrik Stenson.
McDowell made four straight birdies from the second to the fifth. The only blemish on the Northern Irishman's card was a double-bogey on the par-3 11th. Green hit the day's first tee shot at 7:15 a.m. and enjoyed calmer conditions. The Malaysian made five birdies before the turn, and another at the 10th.
Phil Mickelson, McDowell's playing partner, shot a bizarre 66 and trailed by two shots, alongside five others. Mickelson had only five pars while making nine birdies, three bogeys and a double-bogey at the par-4 second hole.
Defending champion Dustin Johnson, who won the inaugural event last year, carded a 67 to be in a group of seven at 3 under. Top-ranked Brooks Koepka and former No. 1 Lee Westwood both shot even-par 70.
Saudi Arabia's controversial image and human rights record affected the lineup at the Royal Greens Golf & Country Club, with Rory McIlroy saying last month the country's image was a factor in the four-time major winner declining his invitation and a reported $2.5 million appearance fee.
Mickelson said last month on Twitter he understood those fans "who are upset or disappointed" about his decision to play.