His opening round included an eagle, eight birdies and a bogey at the par-4 11th hole.
The 63 bettered the single-round course record at the par 72 County Louth Golf Club by one stroke.
"I enjoy links golf and I enjoyed it a lot today," the 26-year-old Molinari said after the best 18 holes of his European Tour career.
His older brother, Edoardo, was U.S. Amateur champion in 2005.
Edfors shot 8-under 64 and held the lead before Molinari finished.
"I didn't actually hole a putt over ten feet all day," Edfors said about his round.
A group of seven, including early leader Paul Lawrie of Scotland, shot 6-under 66 to share third place and 10 others shot 5-under 67 to share fourth place.