Early goals in each half had given the away side the lead, but Bilbao came storming back and might have won it at the end.
After a bright start from both sides, it was Elche who took the lead early on thanks to David Lomban's powerful header.
They could have doubled their lead moments later when Aaron found space outside the box, but his curling effort rebounded off the crossbar.
Gaizka Toquero then produced an acrobatic volley shortly before the break but his spectacular effort was straight at the goalkeeper, while Aaron saw his shot tipped wide.
The Elche man was not to be denied for long, though, collecting a loose ball in the box just after the break and firing into the bottom corner.
The hosts pushed on from their goal with chances for Toquero and Muniain before drawing level with 20 minutes to go. Substitute Aritz Aduriz found space in the box and headed past the goalkeeper just seconds after coming on.
Elche's defence produced two timely blocks in the final five minutes and survived a strong penalty shout to ensure that the visitors took home a point.