The France forward signed a five-year deal after Barca agreed to trigger his €120 million (£107m/$134m) release clause, goal.com reported.
Griezmann was seemingly set to join Barca in the summer of 2018 only to confirm during a television documentary he was staying at Atletico.
The 28-year-old forward signed a new contract with Atletico soon after, but reports surfaced in the second half of the 2018-19 campaign that he was hoping for another chance to join the Liga champion.
And the attacker acknowledged that U-turn in his first public comments since becoming a Barca player on Friday, saying he was taught as a child that opportunities can come around again.
"When I was a boy, my dad taught me that trains don't come around just once," Griezmann said in a video posted on Twitter by Barcelona.
"Now it's time to take on the challenge of a new destination. Finally, our paths cross.
"I'll defend the Barca colors with all my determination and commitment. It's our time."
In addition to last year's documentary and the player's subsequent reversal, the transfer saga has seen Atletico levy accusations of improper conduct at both Barcelona and Griezmann, claiming they had been in negotiations while Diego Simeone's side was still battling in the Champions League.
And despite Friday's announcement by the Liga champion, the saga does not appear to be over, as Atletico insists Barcelona should have paid €200 million to activate his release clause.
Atletico contend the France international agreed terms with Barca before his clause dropped on July 1 and thus the Madrid club is entitled to the €200m figure.
Griezmann's former club released a statement Friday indicating it will take legal action to recoup that fee.
"Atletico Madrid believes the termination of the contract occurred before the end of last season due to events, acts and demonstrations carried out by the player and that is why it [the club] has already started procedures it considers appropriate for the defense of its rights and legitimate interests," said the club's statement.