Griezmann made the announcement during a documentary aired on Thursday, with the film produced by Pique's investment company Kosmos.
France star Griezmann has long been linked with a move away from Atleti, with Barca holding talks with the 27-year-old's representatives in the latter stages of the 2017-18 season.
However, Griezmann - whose brace against Marseille secured a Europa League triumph for Diego Simeone's side in May - has elected to stay put, committing his future to Atleti as they look to secure a Champions League title.
"Another way to see the footballer's life and his doubts to make the most important decision of his career," Pique tweeted.
"Thanks to the team at Kosmos Studios and especially to Griezmann for these last few weeks.
"You decided to stay at Atleti and I hope you go great this season!"
Otra manera de ver la vida del futbolista y sus dudas para tomar la decisión más importante de su carrera. Gracias al equipo de @Kosmos_Studios y en especial a @AntoGriezmann por estas últimas semanas. Decidiste quedarte en el @Atleti y espero que te vaya genial esta temporada!— Gerard Piqué (@3gerardpique) June 14, 2018
His "decision" behind him, Griezmann can now focus on the World Cup with France.
Les Blues begin play in Group C against Australia on June 16.