Saul recognises Costa had to go the extra mile to force a move out of Chelsea last month.
He told Marca: "It was not just about Diego and Atlético. He belonged to another club and if they do not want to sell, there is not much to do.
"You can not spend 300 million, more so at Atlético. The club and Diego were patient. He could have done something else but he wanted to be here.
"We must thank him."