Alex Berenguer put the hosts ahead in the ninth minute and they missed a slew of chances to increase their lead before a Stefan Savic header pulled Atletico level in the 77th.
Diego Simeone's pacesetters went for the winner but were caught out down the other end when Athletic defender Inigo Martinez headed home from a corner in the 86th minute.
"People ask me what the most important factor will be and it's obvious to me that it will be the mental factor," Simeone said after the match.
"We did not play the way we wanted to in the first half, it was totally different in the second, and the team that manages that emotional balance the best will have the best chance of winning the title."
"We're devastated," said Atleti captain Koke. "We looked to get back in the game in the second half by any way possible, we drew level then they got the winner from a set piece.
"We just have to keep going and try to win our next game, we badly need to do that."
Atletico's defeat adds extra intrigue to a thrilling four-way title race, with three points separating them and fourth-placed Sevilla, who beat Granada 2-1, with five games left.
Champions Real Madrid are second on 71 after being held to a 0-0 draw at home to Real Betis on Saturday.
Atletico visit struggling Elche next before a blockbuster game away to Barcelona on May 8.
"We're still fighting for the title and we believe we can win the five games we have left," added Koke.
After a poor run of results let Real, Barca and Sevilla back into a title race, which had looked like Atletico's to lose, Simeone's side had calmed their nerves with back-to-back wins over struggling Eibar and Huesca.
They faced an Athletic side without a win in any competition since March 7 and who lost two Copa del Rey finals this month, being hammered 4-0 by Barcelona in last week's showpiece.
But Athletic looked closer to the energetic and intelligent side seen in their first few weeks under coach Marcelino.
They missed several chances to increase their lead before halftime against a passive and disjointed Atletico.
The changes gave Atleti more urgency and led to them dominating most of the second half but they still lacked a cutting edge and struggled to contain Athletic's counters.