The peloton including all the key players pulled back a break with an elite group of seven riders together at the base of Seskin Hill.
Bevin made the big push up to the leaders before Eibegger made the final run for the win.
An exhausted Eibegger was thrilled to secure his team a second stage win on this year's race after riding out in front for the majority of the race.
Clemens Fankhauser (Austria), who finished 33 seconds back in the peloton, retains the tour leader's yellow jersey with Bevin holding a 10-point advantage in the green points jersey battle with two stages remaining.
Tomorrow's penultimate stage to Baltinglass, on the edge of the Wicklow mountains near Dublin, features seven categorised climbs including two category one ascents ahead of the final flattish stage to the finish in Skerries north of Dublin.