With second-placed Real Madrid facing Villarreal, Diego Simeone's men needed a win to guarantee the title, but their hopes hung in the balance when Oscar Plana put relegation-battling Valladolid ahead.
But Correa's superb solo effort restored parity, with a glaring miss from Shon Weissman at the other end setting the stage for Suarez to complete the turnaround.
Madrid produced a late turnaround to win 2-1 but finished two points behind their city rivals in the table.
The pressure on Atleti's shoulders was reflected in a frantic start from the league leaders, with Suarez and Saul Niguez snatching at efforts before Valladolid struck.
In the 18th minute, a breakdown in Atleti's attack resulted in Marcos de Sousa feeding Plano who, after just keeping himself ahead of a glut of chasing defenders, slotted coolly beyond Jan Oblak.
Jawad El Yamiq nearly handed Atleti a reprieve with what would have been a calamitous own goal, and with Weissman having squandered a chance to double Valladolid's lead, Suarez went close after the restart, only for Saidy Janko to clear the Uruguayan's goal-bound header.
Yet the pressure finally told in the 57th minute.
It looked like the chance to shoot had gone when Correa took a heavy touch, yet nifty footwork created space between four defenders and the Argentine prodded a brilliant finish low into the right-hand corner.
Valladolid seemed all set to restore their lead just after the hour, only for Weissman to again miss a golden opportunity, heading over a gaping goal after Oblak had parried Lucas Olaza's strike.
Atleti made Valladolid pay, Suarez pouncing on Michel's sloppy pass to race clear and curl home the decisive blow, clinching Simeone's second LaLiga crown and condemning the hosts to the second tier.