Atletico scored twice in the first five minutes in Gran Canaria, before netting a third through Koke in the 62nd minute - four minutes after Jonathan Calleri had helped Las Palmas back into the match.
Koke then grabbed his second of the night in the 75th minute, before Thomas Partey made it five late on as the capital outfit picked up their first three points of the season in impressive style.
Alen Halilovic missed out for Las Palmas after picking up a red card against Valencia last weekend. Vitolo was also absent against his parent club, which allowed Momo and Fabio Gonzalez to come into the XI. Former West Ham United forward Calleri, meanwhile, again led the line for the Canary islanders.
Atletico boss Diego Simeone made five changes to the team that started last weekend's 2-2 draw with Girona. Antoine Griezmann missed out through suspension, while Juanfran, Stefan Savic, Gabi and Fernando Torres also all dropped out.
The visitors made the breakthrough inside the opening three minutes, with Correa showing excellent feet in a tight area before poking the ball into the bottom corner as Atletico made the perfect start to the match.
The hosts did settle after a difficult start to the match, but Atletico goalkeeper Jan Oblak remained unmoved as the Canary islanders were restricted to periods of possession with little end product.
Jonathan Viera had Las Palmas' first real chance of the night in the 23rd minute, but the attacker, who had worked space just outside the Atletico box, saw his curling effort saved by Oblak.
Viera then missed the crossbar with a powerful effort just before the 30-minute mark as Las Palmas struggled to make inroads against an Atletico defence that refused to be drawn out of position.
Gabi replaced Luciano Vietto in the 55th minute as Atletico looked to strengthen their grip on the midfield battle, but the change almost resulted in a third goal, with Gabi feeding Carrasco in a dangerous position, only for the Belgian's powerful effort to just whistle wide.
Las Palmas halved the deficit just before the hour, with Calleri rising highest inside the Atletico box to head a wonderful Momo cross into the back of the net.
Atletico, however, firmly closed the door in the 62nd minute when Koke intercepted a poor clearance from Las Palmas centre-back Pedro Bigas before finding the top corner with a brilliant strike.
Koke then completed his brace in the 75th minute with an acrobatic effort after Carrasco's strike had fallen for the Spaniard in a dangerous position.
Viera had the chance to score a second for Las Palmas from the penalty spot in the 81st minute after being felled by Partey inside the box, but the winger saw his spot kick saved by Oblak on what was a miserable night for the team in yellow.
Atletico were not finished, however, and made it 5-1 in the 88th minute when Thomas found the top corner with an effort from just outside the Las Palmas box.