Croatia is the 2018 World Cup runner-up, winning its group that included Argentina before beating Denmark, Russia and England on the way to the final.
The hosts soon increased their lead with two thumping strikes from Marco Asensio, the second of which was given as an own goal to Croatia goalkeeper Lovre Kalinic as it bounced off him after crashing against the crossbar.
Then Rodrigo headed Carvajal's cross early in the 2nd half to make it 4-nil.
That counted as one of four assists for Asensio on the night, with a more conventional one coming four minutes after the restart when he released Rodrigo who nutmegged Kalinic for his fourth goal in six starts for his country.
Isco completed Croatia's heaviest ever defeat with a sixth before the end as Luis Enrique's youthful and exciting side produced a result which will send shockwaves around Europe.
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The two goals are his only two for Spain in 12 appearances to date.
In his press conference on Monday, Enrique said he made a mistake by not initially calling up Aspas for the UEFA Nations League fixtures against England and Croatia.
His first goal was emphatic and he deserved his second effort to go straight in, rather than deflect off the back of the keeper.
"Croatia are the World Cup runners-up and all teams will be motivated against us - and we have to be aware of that".
The centre-back was consistently excellent in Russian Federation but was given a torrid time here.
Rodrigo - who scored the victor at Wembley - beat the offside trap before finishing neatly, Ramos headed in from a corner and Isco turned sharply before firing in from inside the box. This will be the team's first Nations League match.
The 2010 World Cup winners slightly edge Croatia in the head-to-head record with three wins from six outings, while there has been one draw.