Since coming on for Morata after the Real Madrid forward picked up a knock in Spain's 2-0 win over Belgium in Brussels last week, Costa has done nothing but impress new coach Julen Lopetegui.
Lopetegui, who recalled Costa after the Brazilian-born player was omitted from Spain's Euro 2016 squad, was delighted to see the Chelsea striker score twice in the 8-0 demolition of Liechtenstein in Monday's World Cup qualifier.
Costa scored just one goal in 10 appearances for the national team when Vicente del Bosque was coach.
"The more competition there is, the better it is for Spain and for our football," Morata, who impressed at Euro 2016 and also scored twice against Liechtenstein, told Marca.
"We have to continue like this. We have reason to believe in our national team."
Lopetegui elected to start Costa on Monday and then bring Morata on as a substitute.
While Del Bosque's Spain usually played without a clear forward, Lopetegui has shown quickly that he will start with at least one and has demonstrated faith in Costa.
Costa, who has had a bright start to the Premier League season with Chelsea, had not scored for Spain since October 2014 before his brace against Lichtenstein.
"In the end, the players own their destiny," Lopetegui said.
"We try to help them but Diego is a great player and he proved it. It's true that strikers provide much more than just goals. We are happy for him and for Alvaro."
Another player helping Spain get the Lopetegui era off to a promising start is David Silva.
The Manchester City winger scored both goals in the friendly against Belgium and also struck two more against Liechtenstein.
"When one has so many opportunities and good strikers the goals come," Silva said. "It's good to have both our strikers score."
Spain have a major qualifier against Italy on October 6 before playing in Albania three days later.
Asked about the next big meeting with the Azzurri following Italy's 2-0 defeat of Spain in the last 16 at Euro 2016, Lopetegui reckoned the team was looking forward to the challenge.
"Although it's only our second qualifier, they are three important points," he said.
(Editing by Ian Chadband)
