A week after striking twice in a 2-1 victory in the opening round at Celta Vigo, the diminutive Argentinian striker struck again in the 39th minute.
Atletico, who won the title in a season played behind closed doors, finally got to share their trophy with their fans. For the first time in 18 months there was a crowd at the Wanda Metropolitano on Sunday, where 24,926 were allowed in.
The home club also presented summer signing Rodrigo de Paul.
The Argentinian international midfielder, a recent Copa America winner, was willing to help out in defence and assisted on his compatriot's goal, with a pass very like the one with which he set up Angel di Maria for Argentina's winner in the Copa America final against Brazil in July.
De Paul lofted the ball over the visiting defence and into the path of the sprinting Correa.
The striker eluded an untidy challenge on the edge of the penalty area by Elche goalkeeper Kiko Casilla, ran on to the ball, steadied himself and then placed a shot past the backpedalling defenders with the outside of his boot.
"Correa plays where he feels most comfortable," said Atletico coach Diego Simeone.
"When he plays in the centre of the attack, he is very fast, he is vertical, he can turn around like nobody else, he is dangerous because you can't touch him."
Elche dominated possession but Atletico had twice as many strikes on goal.
Luis Suarez again came off the bench for Atletico, replacing Frenchman Thomas Lemar after 63 minutes and picking up a yellow card in the dying seconds.
"He is very important for us," said Simeone.
"He'll get better and better. The preparation for this season has been complicated for everyone, but even more so for us. We mustn't forget that we are the team with the most internationals who were in tournaments this summer, many of them in South America."
Atletico moved to the top of the table but Real Madrid, 4-1 winners over Alaves in the opening round, can retake the lead when they visit Levante in Sunday's late game.
In the afternoon, Real Sociedad, beat promoted Rayo Vallecano 1-0 in San Sebastian.
Spanish international Mikel Oyarzabal scored the only goal with a penalty awarded for a handball by Albanian defender Ivan Balliu, after 68 minutes.