Girona secured a 2-0 victory at Las Palmas which virtually guarantees they will play in the competition for the first time next season, after Athletic's defeat.
Diego Simeone's Atletico, fourth, moved six points clear of their Basque opponents, fifth, with five matches remaining.
After European elimination and then a woeful loss at Alaves last weekend in the top flight, Atletico got back on track with strikes from Rodrigo De Paul and Angel Correa, as well as an Unai Simon own goal.
Nico Williams had levelled De Paul's opener in the first half for Athletic, minutes after suffering alleged racial abuse as he went to take a corner.
Defeat leaves Copa del Rey winners Athletic unlikely to return to the Champions League for the first time in a decade.
"We were really excited for this game, we went for the three points but it was not to be," Nico Williams told DAZN.
"It's been a good season and we'll keep going until the end."
The Spain winger explained he heard "monkey noises" from one area of Atletico's stadium.
"There weren't many of them. There are stupid people everywhere... I hope this changes bit by bit," he added.
Atletico fans greeted visiting Los Leones forward Raul Garcia, retiring at the end of the season, with a tribute. He was part of their squad that won La Liga in 2014.
Inaki Williams hammered an early effort off target for the visitors before Atletico took the lead against the team which has beaten them three times already this season.
A poor clearance fell to the feet of De Paul on the edge of the box and the Argentine fired home with the help of a significant deflection off Ruiz de Galarreta.
The game was paused momentarily after the apparent racial abuse was levelled at Nico Williams when he went to take a corner in the 36th minute.
Moments before half-time, he responded with a goal, finishing calmly after Antoine Griezmann gave the ball away to spark an Athletic counter-attack.
Nico Williams celebrated by tapping his arm in celebration, an apparent reference to his skin colour after the abuse.
"(The celebration) was with a bit of anger, it's not normal to be insulted for the colour of your skin," he added.
Atletico pulled back ahead in the second half when Correa controlled Koke's chipped ball over the top well and then stayed composed to beat Simon.
The hosts made the game safe when Samuel Lino hit the post with a low shot and the ball ricocheted in via the back of goalkeeper Simon.
"We'll try to continue in the same way, game by game, as the season is not over," pledged Atletico coach Simeone.
'Make history'
Earlier David Lopez and top scorer Artem Dovbyk's goals helped Catalan minnows Girona climb provisionally second, above champions Barcelona.
Girona trail league leaders Real Madrid by 13 points, having fallen away in the title race, but now are the same distance ahead of Athletic.
Michel Sanchez's side can secure Champions League qualification if they beat illustrious neighbours Barca next weekend at home, among other scenarios.
Remarkably Girona are only in the Spanish top flight for the fourth season in the club's history.
"(Reaching the Champions League would be historic) not just for the club, but for Spanish football in general," said Lopez.
"What we are doing is insane... we're trying to enjoy it, to do the best we can to make history."
Girona goalkeeper Paulo Gazzaniga saved an early penalty from Sandro Ramirez before his team took the lead after 26 minutes following a penalty of their own for a foul on Eric Garcia.
Although Dovbyk's soft spot kick was saved by Alvaro Valles, Lopez was on hand to smash the rebound into the roof of the net.
Dovbyk doubled Girona's lead in the 57th minute with his 19th goal of the season when he was given another chance from the spot after Yan Couto was fouled.
Barcelona, third, face Valencia on Monday, while runaway league leaders Real Madrid beat Real Sociedad 1-0 on Friday.
