After six defeats in their previous 10 games, Atletico will hope a stirring victory can serve as a springboard for the second half of the season.
Simeone was ecstatic at the final whistle, celebrating with the players and fans like a coach who has had his own position questioned in recent weeks.
"It was a first half in which we weren't ourselves," said Simeone.
"We went into half-time feeling down, anxious, sad. We said if we had to lose, we had to lose with pride, with courage, and that if we scored a goal, the fans would carry us back, which is what happened."
Defeat could have seen Atletico finish the weekend in seventh, with Barcelona, Real Sociedad and Rayo Vallecano all still to play.
Instead, they pull three points clear of La Real, and four ahead of Barca, who play Getafe and Alaves respectively on Sunday.
Valencia, meanwhile, would have been right in the mix for the top four with a win but they sit ninth, seven points behind Atletico in fourth.
The visitors took the lead in the 25th minute when Musah arrowed a superb shot into the bottom corner at the end of a break that began when Hermoso was spun too easily by Goncalo Guedes.
Atletico were even feebler for the second, losing a handful of challenges on the edge of the area as the ball bounced around unclaimed. Toni Lato was allowed to tame it, and slid it through for Duro, who finished.
Joao Felix went off in the second half and it was not clear if the whistles were for the Portuguese's lethargic performance or the decision to take him off.
But his replacement, Cunha, pulled one back, left free to sidefoot in Yannick Carrasco's corner from a yard out.
Atletico swarmed forward. Luis Suarez blazed over, Hermoso headed too high and Correa's dive was tipped wide of the far post.
In the 91st minute, Carrasco surged forward again and while his feathered cross with outside of his foot was just out of reach for Suarez, Jaume Domenech spilled it out to Correa, who fired in.
There were six minutes of injury-time to find a winner and it arrived in the 93rd. Cunha this time was the instigator, collecting, advancing and then shooting towards the back post, where Hermoso was left free with an open net.
Sevilla were unable to complete a dramatic comeback of their own against Celta Vigo as another stumble gives Real Madrid the chance to move six points clear at the top on Sunday.
Celta hit Sevilla twice in three minutes at the end of the first half as Franco Cervi and Iago Aspas gave the visitors a commanding lead at the Ramon Sanchez Pizjuan.
A stunning strike from Papu Gomez gave Sevilla hope and then Oliver Torres equalised four minutes later, leaving the hosts with 16 minutes to find a winner.
But Celta held on for a 2-2 draw that means Sevilla only cut the gap to Real Madrid to three points, with Carlo Ancelotti's side playing their game in hand at home to Elche on Sunday.
It brings an end to a rough week for Sevilla, who drew 1-1 at Valencia in midweek, on the back of a surprise exit at the hands of local rivals Real Betis in the Copa del Rey last weekend.
Their coach Julen Lopetegui has been suffering from Covid symptoms this week and was not on the bench again, with his assistant Pablo Sanz in charge on the line.
"I can't blame these guys for anything because with a bit of luck we would have won," said Sanz.