Floren, who has yet to finish higher than 25th in a tournament this year, had four birdies to go with his eagle at the 338-yard, par-4 13th hole. That left him one stroke ahead of Jean-Baptiste Gonnet, Michael Hoey, Shane Lowry, Joost Luiten and Peter Uihlein.
Another six players, including golf Hall of Fame inductee Jose Maria Olazabal, are tied at 4-under, two strokes behind Floren.
Floren started on the second nine of the Carton House Golf Club and was 2-under through eight holes, with seven pars to go with his eagle. He made the turn at 3-under, however, with an 18th-hole birdie.
Birdies at Nos. 1 and 4 pulled him into a tie with early leader Lowry before a birdie at the 394-yard sixth gave him the outright lead. Floren nearly made it a two-stroke edge when his 30-foot birdie putt at 9 just missed.
Gonnet, Hoey and Lowry each had one bogey and six birdies Thursday but Luiten and Uihlein had much more diverse rounds. Luiten had two bogeys and five birdies in the second nine alone while Uihlein posted a pair of bogeys and four birdies over Nos. 4-9.