The visitors took an early lead through David Lomban, which was doubled early in the second half by Aaron Niguez.
Markel Susaeta provided a quick response from the hosts which put them back in contention and Aritz Aduriz then headed in the equaliser.
There were still 20 minutes for Ernesto Valverde's side to find a winner but no matter how hard they tried, it never arrived.
Elche had the hosts under pressure from the off and it was little surprise when they took the lead.
Lomban rose higher than anyone else to head a corner past Iago Herrerin and into the net with just nine minutes on the clock.
Fidel Chaves then came within inches of scoring what would have been one of the goals of the season. The midfielder had the ball by the left corner of the area and shot with the outside of his foot, curling the ball into the air, but it bounced off the upright and out.
Centre-back Aymeric Laporte came close to pulling Athletic level, glancing a header just a few inches over the bar.
The Frenchman, who scored Athletic's winner against Getafe on Monday, had another chance to score with his head in the second half but again missed the target.
Iker Muniain squandered the best chance of the game less than a minute into the second half. Susaeta sent a ball into the box from the right which bounced off a defender and into the path of the 20-year-old, who was unchallenged but sliced the ball over the bar.
Elche went down the other end to double their lead. It took three Athletic defenders to take the ball off Carles Gil but the players got in each other's way when they tried to clear the danger and instead the ball fell to Aaron, who shot into the bottom corner from inside the box.
The Athletic of last season would have given up in the circumstances, but Valverde's side are now made of sterner stuff.
Susaeta put them back into the game eight minutes later, getting on the end of a fine cross from Muniain to sidefoot the ball home.
With 21 minutes remaining, Athletic completed the comeback. Benat delivered an outswinging corner and Aduriz met the cross with his head to score his second goal in three games.
Once level, the hosts went searching for the winner. Benat saw a goalbound shot blocked by the flailing leg of Elche defender Alvaro Botia and then Gaizka Toquero, for the second time in the game, failed to hit the target at the near post.