Real Madrid and Barcelona were the only sides to take all three points from this part of the Basque Country last season, but Atleti upped things after the restart to get the job done.
Raul Garcia did manage to pull one back for the hosts against his former side in added time, yet it proved to be too little, too late as Atletico picked up successive league wins for the first time this season.
Both goalkeepers were in inspired form in a lively first half, with Kepa Arrizabalaga - strongly linked with a free transfer to Real Madrid at the end of the season - doing well to keep out a deflected free kick from Antoine Griezmann.
Aritz Aduriz nearly got on the end of Unai Nunez's knockdown, but he could only get the slightest of touches to the ball, while Mikel San Jose comically miscued when trying his luck from inside the box.
Atletico were having to do plenty of defending but they came within inches of opening the scoring 37 minutes in after Nicolas Gaitan pounced on Nunez's error, charging through on goal but picking out the post from a slight angle.
Kepa had to keep out a near-post attempt from Griezmann a couple of minutes later to ensure that the score remained level, though it was Athletic who had more reason to be upset about the goalless nature of the first half.
After the referee pointed to the spot for Filipe Luis's challenge on Garcia on the 18-yard line, which proved to be the wrong call from the officials, Aduriz stepped up to take the penalty but saw his effort kept out by set-piece specialist Oblak.
A chance also came and went for Filipe Luis right at the end of the first half, getting on the end of a Griezmann pass and putting his boot through the ball to call Kepa into action once again.
Atletico started the second half much the better side, coming close through another on-target Griezmann attempt, and they were rewarded 55 minutes in when Correa bagged his third goal of the campaign.
The goal was created by Griezmann and Koke, who linked up for the latter to square to Correa a few yards out for the simplest of finishes to finally get the beating of Kepa.
Diego Simeone was not a happy figure on the touchline, however, as for the second time in the match Griezmann had a goal ruled out, before seeing the French striker have a penalty appeal turned down.
Los Rojiblancos were in complete control as the hour mark arrived, as Griezmann and Correa continued their battle with young keeper Kepa.
The scoreline remained slender heading into the final quarter of the match, though, which almost told when Garcia volleyed over the bar from a glorious position.
Substitute Carrasco appeared to make sure of the points 73 minutes in when latching on to a lofted Griezmann pass and tucking past Kepa with a composed finish.
Oblak was finally beaten in the closing stages as Garcia blasted home from close range, but the visitors saw through the remaining seconds to close the gap on league leaders Barcelona.