France forward Griezmann, 27, pledged his long-term future to Atleti in June, signing a contract extension through to 2023.
He had been widely tipped to join Barca after a protracted transfer saga that remained unresolved until the early stages of the World Cup.
Costa, who voiced regret over leaving for Chelsea in 2014, believes his team-mate was right to spurn the move in favour of continuity.
"I understood Griezmann. People criticise and people around him may have told him that he needed a change, just as what happened to me," the Spain striker told Marca.
"Outsiders were very opinionated, but we both are very guided by our families.
"He stayed true to his heart, and he wanted to stay, to be here. From the beginning I knew [he] wanted to stay.
"Of course, an offer from Barcelona is important. [With Lionel] Messi, the club...it's normal to be excited. Atletico are a big club too, and Griezmann is a leader.
"He stayed and gave us the Europa League. Then he won the Super Cup and the World Cup. He's a boy who showed the feelings he had.
"This is the strength that the club has. We know that Atletico can fight with any other club."
Atleti, third in LaLiga after six matches, could climb above Real Madrid when the city rivals meet at Santiago Bernabeu Stadium on Saturday.