The Spanish champions were beaten 3-2 by Olympiakos in the UEFA Champions League on Tuesday, and dropped points in La Liga mean they could finish the weekend four points adrift of Barcelona, should the table-toppers beat Levante on Sunday.
Pablo Hernandez gave Celta the lead with an outrageous finish after 19 minutes, but Atletico were ahead at the break courtesy of goals from centre-back pairing Miranda and Diego Godin.
However, the former undid his good work by giving away a penalty early in the second half, which was subsequently converted by Nolito.
Atletico piled the pressure on in the latter stages, forcing Celta goalkeeper Sergio Alvarez into several saves before substitute Alessio Cerci had an injury-time goal disallowed for offside, but they had to settle for a point in the final game of coach Diego Simeone's touchline ban.
The hosts almost broke the deadlock in the 15th minute as Tiago burst into the area to meet Juanfran's low cross, but the midfielder's first-time effort flew just wide.
And Atletico were made to regret that miss four minutes later, as Hernandez gave Celta the lead in audacious fashion.
The Chilean tussled with Godin as Carles Planas' cross from deep appeared set to fly over their heads, but Hernandez somehow managed to pull off an incredible volleyed backheel, sending the ball past Miguel Angel Moya.
Alvarez then twice denied Atletico an equaliser, but he was powerless in the 31st minute, as the unmarked Miranda directed Koke's free-kick into the net.
And the champions completed the turnaround just before the break as Godin met Gabi's corner with a bullet header, giving Alvarez no chance.
Celta started the second half in lively fashion, and almost drew level early on as Fabian Orellana played Hugo Mallo into the area only for the right-back to fire straight at Moya.
They did restore parity in the 53rd minute, though.
Planas produced a delightful turn in the area to beat Miranda and the Brazilian clumsily tripped the former Barcelona defender, with Nolito - a half-time substitute - duly converting the resulting penalty.
Simeone's team should have stolen all three points in the 78th minute as Raul Jimenez, starting in place of the injured Mario Mandzukic, pounced on a loose ball before firing straight at Alvarez from eight yards.
Cerci made his Liga bow off the bench and almost enjoyed a dream domestic debut when he fired beyond Alvarez from close range in injury time, only to be denied by the assistant referee as Atletico were left frustrated.
