Griezmann scored two late goals as Barcelona came from behind to beat third-division club Ibiza 2-1 in the competition's round of 32 on Wednesday.
Days after their heartbreaking exit from the Champions League, Barcelona received another setback with an injury to striker Luis Suarez.
Valencia won 1-0 at home to reach the Copa final for the first time in more than a decade, where it will play Barcelona on May 25.
Luis Suarez scored twice as Barcelona advanced 4-1 on aggregate after being held to a 1-1 draw in the first leg of the semifinals at Camp Nou.
The two Spanish clubs are to play each other twice within a week, starting with the second leg of the Copa del Rey semi-final on Wednesday at Santiago Bernabeu Stadium.
Messi regained his scoring touch just before the two decisive matches against Real Madrid.
Santiago Solari said Real Madrid would have a day less to prepare than its rival Barcelona ahead of their Copa del Rey El Clasico.
Lucas Vazquez gave Real Madrid the lead in the 6th minute before Malcom equalised for the hosts in the 57th minute to earn a draw at the Camp Nou.
With Barca quiet on Messi's injury, Madrid arrive for the first-leg match at Camp Nou enjoying its best period of the season and with nearly all of its players available.
The first leg will be next week in Barcelona, with the return match in Madrid on February 27, just before their Spanish league game at Santiago Bernabeu Stadium on March 3.
Real Madrid advanced 7-3 on aggregate having won the first leg 4-2 at Santiago Bernabeu Stadium last week.
Phillipe Coutinho netted two goals in his best game of an up-and-down season for the Catalan club, while Mess and Suarez scored in the latter end to complete the demolition.
Sergio Ramos twice gave Madrid the lead in a 4-2 win against a gritty Girona team that had opened the scoring less than 10 minute into the match at Santiago Bernabeu.
It was the first time Barcelona failed to score since a 3-0 loss to Roma that eliminated the team from the Champions League last season, 39 matches ago.
Barcelona has made it at least to the quarterfinals in the Copa in nine straight seasons. It won the title five times in the last seven years.
Atletico's usually stout defence struggled in a 3-3 draw and allowed an 88th-minute equalizer that kept Simeone's team from reaching the last eight for the first time since the coach took over in 2012.
The hosts kept threatening throughout the match, taking advantage of a Barcelona defence that was without Gerard Pique and Jordi Alba.
Sergio Ramos netted his 100th career goal to put Madrid ahead, and Lucas Vazquez and Vinicius Junior added to the lead.
Both Marco Asensio and Isco scored two each as Real Madrid thrashed Melilla 6-1 in the Copa del Rey.
Barcelona needed an injury-time goal to win 1-0 at Cultural Leonesa, while Madrid cruised to a 4-0 victory over third-tier Melilla in a first-leg match in the round of 32.
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