Griezmann came off the bench in the second half and turned the tie in Atletico's favour, teeing up Angel Correa and then scoring himself to give his team a 3-2 lead at a rocking Wanda Metropolitano.
But Girona hit back one more time to go through to the quarter-finals as Seydou Doumbia's shot found the corner in the 88th minute, snatching a 4-4 draw on aggregate after last week's first leg finished 1-1.
Griezmann had scored then too, and in between against Levante on Saturday. This was his eighth goal in as many matches.
The Frenchman, however, had started on the bench while Atletico were punished for some uncharacteristically lacklustre defending.
A sumptuous volley from Valery Fernandez and Cristhian Stuani's header put Girona in charge after Nikola Kalinic had put Atletico in front.
Perhaps Diego Simeone had one eye on the Champions League tie against Juventus next month, not to mention his side's title challenge, which has them five points behind Barcelona and the Catalans' closest challengers in La Liga.
But it is now six seasons since Atletico last reached the final of Spain's most prestigious cup competition, which represented their most likely source of a trophy this term.
They had looked set for a comfortable night when Kalinic was allowed to drift in behind Girona's dozing defence in the 12th minute, let the ball bounce and drive in on the half-volley.
Instead, Girona levelled thanks to a superb volley from Valery. The cross was flicked to the back post, where Valery cushioned the ball with his chest, before hammering it into the far corner.
Correa struck the post after half-time, his shot squirming through the hands of a relieved Gorka Iraizoz, while Saul Niguez hobbled off, replaced by Rodrigo.
Girona pulled ahead. Atletico were uncharacteristically generous in giving Stuani space at a free-kick and he nodded in Aleix Garcia's cross with ease.
The ball had hit the net in the 59th minute and in the 61st, Griezmann entered the field. In the 65th, he set up a goal too, Kalinic letting the Frenchman's pass run but doing enough to distract Girona's defence. Correa nipped in and finished.
With Girona still progressing on away goals, Atletico had just over 20 minutes to find a winner. Santiago Arias thought he had it but VAR showed him just offside. So did Griezmann, as he latched onto Thomas Lemar's scoped ball through and smashed it into the roof of the net.
He celebrated like he had won the tie but there was one more sting in the tail. Borja Garcia raced clear down the right and his cutback landed at the feet of Doumbia. His shot flew through the legs of Arias and snuck into the corner.