Atleti leapfrogged Rayo in the standings and are now fourth with 32 points from 19 matches. Rayo are fifth, with 30 points. Real Madrid, who lost at Getafe earlier, lead LaLiga with 46 points from 20 games.
Correa, playing out of position as a pure striker in the absence of Antoine Griezmann and Joao Felix, who both missed Atletico's first game of 2022 due to positive COVID-19 tests, led Atletico to a victory where they dominated from start to finish.
"After four consecutive losses, we needed this win," man-of-the-match Correa told reporters.
"It's a great way to start the year because we won playing well, dominating; we are really happy with the attitude and the result."
The win was a relief for Atletico manager Diego Simeone, who had never previously lost more than two successive league matches since taking charge in 2011.
"It was tremendously important [to get the win]. The players showed it with how they approached the game, from the first minute to the last, with a lot of heart. We know that results are the reality but in the four previous games we'd done important things well too, and the mistakes had hurt us more than we deserved," Simeone said.
Simeone had tested positive for COVID-19 earlier in the week but was cleared to take his place on the sidelines early on Sunday after his latest test came back negative.
Missing Griezmann, Felix and captain Koke due to the pandemic and Spanish international flanker Marcos Llorente because of injury, the Argentinian coach decided to change his system from three defenders to a classic 4-4-2.
The change was key as defenders Kieran Trippier and Renan Lodi dominated on the touchline, winning in every phase of the game.
The first goal started with a combination between the English right back and midfielder Rodrigo de Paul, who passed to Carrasco who entered the area and shot at the goal. When the ball was deflected, Correa was the quickest to get to it and open the scoring.
In the second half, Brazilian international Lodi sprinted up front and found Correa in the box with a perfect cross.
Simeone added: "Angel should be highlighted for his finshing with the goals. He's a forward and having [Luis] Suarez, Cunha, Griezmann, Joao [Felix], often we use him elsewhere so that we have an extra player who can win games like he does, as he did last season when we were champions."