Their team-mate Adrian Lopez, alongside Betis' Benat, also make their first appearances for La Furia Roja, with all four hoping to be named in Vicente del Bosque's final 23-man squad for Euro 2012, with the experienced trainer set to name his list on Sunday.
Speaking after the straightforward victory over the Balkan nation, Dominguez told Marca: "I am very happy because it's not every day you make your debut for the best team in the world. I want to thank the coach."
Talking about his chances of going to Poland and Ukraine, the centre-back believes the fact he made the provisional squad means he has hope.
"It's complicated but I am here because I have a chance," he said. "It is a pleasure to share a dressing room with these players and I'm just grateful to be here."
Dominguez's club team-mate, Juanfran, started the game but the defender is not getting too carried away in terms of whether he will make the final cut.
"[I need to] be positive, think as if I'm going to be [in the squad]. We will have to wait for the decision of the coach and whatever decision he makes, we need to respect," the former Real Madrid youth player mused.
"I'm happy to make my debut. I felt very good and enjoyed it with my team-mates. Whether I'm with Atletico or Spain, I try and do my best."
Both players helped Atleti to a fifth-place finish in La Liga, while the duo lifted the Europa League trophy after a 3-0 victory in the final versus Athletic Bilbao.