The Brazil-born striker has been prolific at club level since joining Chelsea, but a first goal at international level for his adopted country proved to be elusive until Sunday.
Costa ended his Spain goal drought as the European champions responded to their shock 2-1 defeat against Slovakia last week with a comfortable Euro 2016 qualifying Group C away victory.
Former Atletico Madrid frontman Costa, who scored Spain's third goal, hopes his strike was the sign of things to come on the international stage.
"It was pretty annoying not being able to score. It bothered me much. Hopefully things go well for me now," he told TVE.
"Every game is a new opportunity to show what I am, I have to be. Hopefully, from now on, things are going very well me."
David Silva, Paco Alcarer and debutant Juan Bernat were also on target for Spain as they proved to be too strong for Luxembourg.
Vicente del Bosque's side are now second in the group, three points behind Slovakia after they won 3-1 in Belarus to maintain their 100 per cent record.
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