Simeone's side leapfrogged Real Sociedad into fourth on 42 points, while Sevilla stayed third in the standings on 46 after beating Las Palmas 1-0, also on Sunday, thanks to a late strike from substitute Joaquin Correa
Antoine Griezmann scored a late winner to complete a stunning fightback by Atletico Madrid, who beat Celta Vigo 3-2 on Sunday with two goals in three minutes.
Atletico looked set to stumble to defeat when John Guidetti put the visitors ahead in the 78th minute for the second time in the game.
But Yannick Carrasco drew the home side level with a stunning volley in the 86th minute and France international Griezmann completed the comeback by sidefooting Kevin Gameiro's flicked header in at the near post in the 89th.
"It was a thrilling and emotional game with a lot of errors between two teams that were looking to win," said Atletico coach Diego Simeone.
"Celta did very well, they made it very difficult on the counter attack but once again we reinvented ourselves in the final 10 minutes."
The home side got off to a nightmare start as Miguel Angel Moya came out to claim a corner but instead tipped the ball onto the head of Gustavo Cabral, who nodded into the unguarded net.
But Fernando Torres quickly pulled them level with a splendid strike, receiving the ball with his back to goal and flicking it over his head into the top corner.
The former Liverpool and Chelsea striker had a strong penalty appeal waved away shortly afterwards but then failed to take advantage when Carrasco was felled by Facundo Roncaglia. It was the sixth miss from Atletico's nine penalties this season.
Simeone's side leapfrogged Real Sociedad into fourth on 42 points, while Sevilla stayed third in the standings on 46 after beating Las Palmas 1-0, also on Sunday, thanks to a late strike from substitute Joaquin Correa.
Real Madrid top the standings on 49 points after winning 3-1 at Osasuna on Saturday, and have played two games fewer than their title rivals. Barcelona are second on 48 after thrashing Alaves 6-0.
Sevilla's Argentine midfielder Correa broke the deadlock in the 80th minute just moments after coming on. He latched onto a long punt from goalkeeper Sergio Rico that was flicked on by Vicente Iborra.
Sevilla, who had failed to win on their previous two outings, inflicted a first home league defeat of the season on Las Palmas, who were thwarted by a couple of outstanding saves from Rico in an end-to-end second half.
"We played with fury and we got a vital result at a difficult ground," Sevilla coach Jorge Sampaoli told reporters.
"Sergio made a couple of incredible saves and it was a very difficult game, but I feel satisfied that the team could stand up and be counted in an important game."