Griezmann made the announcement during a documentary aired on Friday (AEST), 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 2017-2018 season.
However, Griezmann - whose brace against Marseille secured a UEFA 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 UEFA Champions League title.
Writing on Twitter, Pique - likely to start in Spain's World Cup opener against Portugal on Friday - thanked Griezmann for letting the documentary team witness his decision.
"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!"