HILTON HEAD ISLAND, 17 Apr 2015 (AFP) - McDowell, the 2013 champion, has made three cuts in his last six worldwide starts and his best finish was a share of 36th in Malaysia.
Every collected his second-straight Arnold Palmer Invitational crown in his only top-25 finish this season at Bay Hill.
Newly-crowned Masters winner Spieth struggled throughout his opening round and posted a three-over 74, which left him tied for 93rd place.
Spieth posted a double-bogey, two bogeys and a birdie in his round at the Harbour Town Golf Links course.
"I didn't drive the ball well, I didn't particularly strike my irons well, and chipping and putting wasn't there. Just was an off day," Spieth said.
South Korea's Bae Sang-Moon kicked off his week with a four-under 67 and is alone in third place.
Defending champion Matt Kuchar shot a three-under 68. He was joined in fourth place by Kevin Kisner, Morgan Hoffmann, Scott Langley and Cameron Smith.
Northern Ireland's McDowell quickly moved atop the leaderboard early on the back nine.
He drained a 16-foot birdie putt at the 10th. After a par on 11, McDowell moved into the lead thanks to back-to-back birdies at 12 and 13. He parred the last five holes to grab the co-lead at five under.
"I drove it in the fairway, which I haven't been doing enough of this year, and I just felt really good on the greens," said McDowell.
"You have to stay patient. You have to position the ball well at all times, and you have to really be aware of the wind on this golf course."
Johnson Wagner shot two-under 69 and shares ninth with a large group that also includes, Pat Perez, Ian Poulter and former Open Champion Louis Oosthuizen. Wagner recently lost in a playoff at the Houston Open.
Leading first round scores here on Thursday at the Heritage tournament:
66 - Graeme McDowell (NIR), Matt Every
67 - Sang-Moon Bae (KOR)
68 - Matt Kuchar, Kevin Kisner, Morgan Hoffmann, Scott Langley, Cameron Smith (AUS)
69 - Pat Perez, Ian Poulter (ENG), Louis Oosthuizen (RSA), Bo Van Pelt, Ricky Barnes, Johnson Wagner, Graham DeLaet (CAN), John Merrick, Joost Luiten (NED), Jim Renner, Charlie Beljan, Troy Merritt, Ben Martin