The reigning champion ended a four-game losing streak and leapfrogged its opponent, which failed to register a single shot on target at Wanda Metropolitano, into fourth place.
With Diego Simeone on the touchline having received authorisation from LaLiga after a recent positive COVID-19 test, Correa put the hosts ahead in the 28th minute with his first league goal since 29 November (AEDT).
The Argentina international doubled his tally and secured the points when he rounded off a neat move early in the second half.
Despite its poor form, Atletico was searching for a sixth successive win over Vallecano and an eighth in the league on home soil.
The hosts squandered a great opportunity to take the lead in the 23rd minute when Luis Suarez robbed Esteban Saveljich of possession, but sent his attempted chip wide with only the goalkeeper to beat.
But they were not to be denied five minutes later. Mario Hermoso's blocked effort from a Yannick Carrasco cutback ricocheted kindly into the path of Correa, who drilled home through the legs of Luca Zidane.
Correa doubled the advantage within eight minutes of the restart, providing the finishing touch to Renan Lodi's cross from close range after a sweeping move by Atleti.
Thomas Lemar and Carrasco went close to increasing the margin of victory – the latter hitting the post for Simeone's side, who have now kept five clean sheets on the spin in this fixture.