The Atletico Madrid striker, who has switched loyalties from Brazil to la Roja, is set to make his debut for Vicente del Bosque's men in their friendly at home to Italy on Wednesday.
Xavi dispelled any notions that the in-form Rojiblancos forward - whom Del Bosque defended following criticism in the Italian press about his switched allegiances on Tuesday - is taking his time adjusting to his new international surroundings.
"Costa is very happy, he is glad, very happy," Xavi told RTVE. "He has quickly integrated within the group, he's a very cheerful and funny guy.
"He's one more addition to our group and he will do very well at the disposal of this coach and team."
Xavi, who is set to feature at his fourth World Cup in the summer, reiterated that the tournament in Brazil will be the last inter-continental one he takes part in but his undecided whether he will make himself available for Euro 2016.
"I have to put some thought into whether I will be at the next European Championship," the 35-year-old mused.
"I am sure I will not be at the next World Cup, but we'll see what happens before that. We'll see how it goes.
"I'm thrilled to be set to compete in Brazil, though; we expect to do well there and put in a good performance in our friendly with Italy on Wednesday."
The last time the Azzurri faced Spain they were soundly beaten 4-0 in the final of Euro 2012, although they had held the reigning world champions to a 1-1 draw earlier in the tournament.