The home side fell behind after nine minutes when David Lomban headed home from close range, with that goal all that separated the two sides at the interval.
Bilbao were level with 20 minutes remaining when Aritz Aduriz's header found the back of the net, but they could not find a winner as the points were shared.
Read how it all unfolded in Sports Mole's minute-by-minute live commentary of the action below.
ATHLETIC BILBAO: Iago; Iraola, Balenziaga, Laporte, Gurpegi; Susaeta, Iturraspe, Rico, Herrera; Muniain, Toquero
ELCHE: Herrera; Cisma, Lomban, Suarez, Botia; Sanchez, Niguez, Perez; Fidel, Coro, Gil