Suarez came on in the 70th minute at Wanda Metropolitano with Atleti 3-0 up against Granada thanks to goals from Diego Costa, Angel Correa and Joao Felix, while Saul Niguez had missed a penalty.
A stunning cameo followed, with Suarez netting a brilliant double after providing an assist for his fellow substitute Marcos Llorente.
The 33-year-old was deemed surplus to requirements at Barcelona as the club looks to rebuild under Ronald Koeman, but Suarez believes the time was right for him to move on.
"Very good, very happy for the debut, for the three points," he said. "When you make your debut and help the team with goals, you are always happy.
"It is a moment in which I needed fresh air, it is important to change and accept it.
"I come to a club that has given me a spectacular welcome, the atmosphere is very good from the beginning and that is reflected on the field."
Before Suarez stole the show, Joao Felix had put in an impressive display partnering Costa up front, having teed up Correa's goal prior to getting on the scoresheet himself.
"We're feeling good about ourselves. We've completed a great game, scoring many goals and creating lots of chances," he said.
"We have to continue like this. I'm feeling good. I'm happy and enjoying myself. Things are going well.
"This was just the first game, but we want to complete a very, very good season. We're working towards that. We started today and have to continue against Huesca."