Fernando Llorente made his first start of the season for Bilbao since refusing to extend his contract with his boyhood club beyond next summer.
But even with the Spain striker on the pitch with a first-choice side, Bilbao was outplayed by the fellow Basque club.
The 23-time cup winner Bilbao showed none of the verve that led it to last year's final. Eibar, meanwhile, created three clear scoring chances in the first half that its forwards failed to finish.
"We played poorly in the first half and should have been losing by halftime," said Bilbao coach Marcelo Bielsa, whose team has only won two of its first nine games in the Spanish league and is winless in three games in its Europa League group.
"It may sound strange to hear â€" I'm not especially disappointed with drawing with Eibar. But it would be arrogant to say that we showed we were better than our rival."
Bilbao was not the only favorite to flop in the opening leg of the knockout series on Thursday. Fellow first-division sides Levante, Real Sociedad and Osasuna all lost to lower-division opponents.
Levante will have to look to bounce back in the second leg from a 1-0 loss at third-tier Melilla.
Melilla midfielder Fernando Velasco scored from the penalty spot in the 39th after being fouled by Nikolaos Karampelas. Melilla, the team of one of Spain's two enclave cities on the north African coast, held on for the victory despite Carlos David receiving his second booking in the 71st.
Cordoba, playing in the second division, beat visiting Sociedad 2-0 at Arcangel stadium with goals by Javier Patino and Vincenzo Rennella.
Sporting Gijonâ€" relegated from the first division last seasonâ€" edged Osasuna 1-0 thanks to Gaston Sangoy's goal in the fifth minute.
Getafe was the only first-division team to get a win over a lower-division opponent. The Madrid-based team routed second-division Ponferradina 4-0, with two goals from Jaime Gavilan and one apiece for Pablo Sarrabia and Paco Alcacer.
Sevilla, a cup winner in 2010, beat an Espanyol side full of reserves 3-1 to take commanding lead in their tie.
Espanyol looked set to snatch a valuable away draw when Cristian Alfonso canceled out Federico Fazio's early opener with a long-range shot in the 69th.
But Sevilla defender Juan "Cala" Torres volleyed in a corner seven minutes from time before Alvaro Negredo converted a spot kick in added time after Ernest Galan pulled the striker down in the box.
Elsewhere, Alberto Bueno scored five minutes from halftime to give 10-man Valladolid a 1-0 home win over Real Betis.
Valladolid's Henrique Moreno was red carded for striking an opposing player, but Betis failed to take advantage of its extra man for the final 25 minutes.
"I'm happy for the goal because it gives us an advantage heading to Seville, even though we know the series is still wide open," Bueno said.
Mallorca visits Deportivo La Coruna later.
The return legs of the round-of-32â€" the first with Spain's 20 first-division teams joining the lower-division survivors from the opening three roundsâ€" will be played during the last week of November.
