Antoine Griezmann’s stunning first-half free-kick put Atletico in front.
However, Atletico felt the after effects of their efforts in seeing off PSV Eindhoven after extra-time and penalties in the Champions League in midweek as Antonio Sanabria and Carlos Castro turned the game around in the final stages.
Barcelona can now move 11 points clear at the top when they travel to Villarreal on Sunday.
“We were always clear that the objective of the club is to be in the top three,” said Atletico manager Diego Simeone.
“That is what we are working towards. This defeat goes against any aspiration to be champions.”
Simeone named a youthful side with an average age of 23 as he made five changes from the team that started against PSV on Tuesday.
Atletico have often felt they have been prejudiced by the scheduling of their La Liga fixtures around European games this season.
And Simeone hinted at his frustration at having to play early on Saturday, whilst Barcelona and Real Madrid aren’t in action till Sunday.
“I don’t like to look for excuses. I’m happy that we are premium, we play at four o’clock and they see us in China,” he added ironically.
Griezmann retained his place up front and was once again Atletico’s only serious goal threat as he opened the scoring just before the half hour mark.
Sporting had reason to feel hard done by as Pablo Perez looked to have taken the ball before making contact with Saul Niguez on the edge of the area.
And there was no stopping Griezmann’s effort from the resulting free-kick as he curled the ball over the wall and into the top corner.
Simeone looked to hold out for the three points as he withdrew his three most attacking players in Griezmann, Angel Correa and Luciano Vietto after the break.
However, for once, Atletico’s defensive resilience let them down.
Sanabria fired a warning shot when he struck the post from a free-kick 12 minutes from time.
But presented with a similar opportunity a minute later, the Paraguayan rifled a free-kick through the Atletico wall to leave Jan Oblak unsighted.
Castro missed a glorious chance to make it 2-1 four minutes from time when he smashed Sanabria’s unselfish square ball off the bar in front of an open goal.
Of further concern for Atletico was that opportunity came about as Jose Maria Gimenez pulled up with a hamstring injury making him a doubt for the first-leg of their Champions League quarter-final away to Barcelona on April 5.
With Simeone having made all three substitutions, the visitors had to see out the final few minutes with 10 men.
And they couldn’t do it as Sanabria again burst down the left and this time his low cross was converted at the far post by the relieved Castro.
Victory brought Sporting to within a point of survival as Granada and Rayo Vallecano stayed just out of the drop zone with a 2-2 draw.
Youssef El Arabi twice put Granada in front, but Pablo Hernandez and Ze Castro replied for the Madrid side.
Levante remain rooted to the bottom, four points off safety, as goalkeeper Diego Marino’s unfortunate own goal handed Deportivo la Coruna a 2-1 win - their first in 2016.
Las Palmas continued their march towards safety thanks to Willian Jose’s solitary to secure a fourth win in five games, 1-0 away to Real Sociedad.
Malaga moved up to eighth with Ignacio Camacho’s late winner in a 1-0 win at Real Betis.
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