Hierro was interim manager for the national team during the tournament, a last-minute replacement after Julen Lopetegui was sacked.
A team with a history of success, a talented squad and a distinctive style of playing, had just failed to win a trophy for the first time in six years.
Spanish football federation (RFEF) confirmed the development on Sunday afternoon.
Enrique, who played for Real Madrid and Barcelona, won the La Liga, Spanish Cup and Champions League treble as Barca manager in 2015.
Rubiales had earlier on Monday said that the new head coach of Spain's national team needed to be a person of strong character.
Roberto Martinez, who has led Belgium to the semi-finals of the World Cup, is among the coaches to have been linked with taking the Spain role.
The 48-year-old has become La Roja's new coach following caretaker boss Fernando Hierro's decision to stand down from the position on the back of a disappointing World Cup.
"We didn't speak to anyone else, we only spoke to Luis Enrique", Rubiales said. Hierro, who had previously held that role, decided not to return to the post following the World Cup.
Molina, 47, a former goalkeeper with Atletico Madrid, Deportivo La Coruna and Villarreal, earned nine Spain caps as a player.