Simeone's side appeared destined for a third consecutive defeat in all competitions after Yunus Musah and Hugo Duro had put the visitor two goals up at the interval.
Matheus Cunha reduced the deficit in the 64th minute, before added-time strikes from Angel Correa and Mario Hermoso secured a remarkable victory.
“At half time I spoke to the players, I told them what I felt and I have transmitted in recent years, 'if we have to lose we lose with pride'," Simeone said. “The team left their soul out there, they left their lives on the pitch.
"There are nights that remain with you and this one is going to stay with us.
"It was a fantastic second half. We were ourselves. Watching that second half, I felt: this is us."
Atleti had collected just four points from the previous 18 available in LaLiga, with the win boosting its lagging top-four aspirations.
Simeone was almost reduced to tears on the touchline after Hermoso's winner as the defending Spanish champion firmed up fourth spot, three points clear of fifth-placed Real Sociedad.
"What did I feel? That it's marvellous, this place I'm in," Simeone said. "Whenever it feels like it might be going, I want to hold on even tighter. In the first half, my heart hurt."
The result saw Atleti become just the second LaLiga team in the 21st century to come from behind to win a game with two goals in added time, after Betis did so – also against Valencia - in December 2011.