Gore, starting on the back nine at the Bay Hill golf course, combined four birdies with one bogey, 11 pars, and an eagle at the par-5 12th hole.
Australia's Stuart Appleby leads a group of six at 3-under 67, tied for fourth.
Defending champion and five-time invitational winner, Tiger Woods, shot 2-under 68, joining six others in a tie for 10th place.
"I held it together with some up and downs towards the end of the front nine," Woods said.
One player with Woods is Rocco Mediate, who lost an epic 19-hole playoff to him at last year's U.S. Open in La Jolla, Calif.
Palmer's event is two weeks later on the schedule this season and likely will be the final test for some in preparation for 2009's first major, next month's Masters at Augusta (Ga.) National.
Swim Week Miami Beach 2014 [PHOTOS]