Atletico Madrid had 19 shots and, unusually for them, dominated possession.
Russia keeper Igor Akinfeev saved Koke's and Iago Aspas' spot kicks in their first ever penalty shootout to send the hosts through.
Koke, Atletico Madrid's dependable midfielder, has quashed speculation of him seeking a transfer to Chelsea and has pledged his services to Atletico next season.
Spain crumbled to a humiliating 5-1 loss against the Netherlands in the FIFA World Cup, last June. The Spanish never recovered from the thrashing and failed to make the knockouts.
Atletico Madrid midfielder Koke suffered a torn hamstring in his side's 4-0 thrashing of Real Madrid and is likely to be ruled out for over a month.
EPL giants Manchester United and Arsenal are believed to be interested in the playmaker. But Koke, approaching his 200th club appearance ahead of his 23rd birthday, said he wants to remain at the club he has supported since his childhood.