Griezmann scored twice, including the 40th-minute opener, but his goals were cancelled out on both occasions at Estadio de La Cartuja.
Athletic's Oscar de Marcos equalized in the 42nd minute before substitute team-mate Asier Villalibre forced extra time at the death, after Griezmann had restored Barca's lead with 13 minutes remaining.
Inaki Williams then scored a sublime goal in the third minute of extra time to secure Athletic's third Supercopa crown and first since 2015, with Messi's red card for lashing out at Villalibre adding to Barca's woes in the dying moments.
"How do you think we are?" Griezmann replied when asked how Barca felt post-game.
"We're screwed, angry, upset. When you lose a final, these are all the bad feelings you have.
"Athletic pressed very well. They gave everything until the last minute, and then what happened, happened. We have made mistakes in our strategy and we'll have to work on it during the week.
"We didn't defend well, we didn't communicate. When the ball comes out someone needs to shout."
France star Griezmann has scored five goals in his past four finals in all competitions for club and country.
He has also been directly involved in seven goals in his past four appearances for Barca across all competitions, after failing to score or assist a goal in any of his previous seven matches.