Suarez, 33, joined Atletico last month after Barca told him he was no longer part of their plans despite having a year remaining on his contract.
The Uruguay international was one of several big-earners who moved on this summer as Barca attempted to reduce their wage bill to limit the financial repercussions of the coronavirus pandemic.
"There were other ways to tell me that they wanted to change things," Suarez, who was informed he was surplus to requirements in a brief telephone conversation with new coach Ronald Koeman in August, said.
"The way it was done and the feeling that they're kicking you out is what hurt the most. Those few days [before my Atletico debut on Sept. 27] were really hard.
"I cried because of everything I was going through. I didn't take the message from the club about moving on well. More than anything because of the way they handled it, because you have to accept when a cycle has come to the end."
Barca captain Lionel Messi was also angry at the way Suarez's departure was handled. He posted a message on social media when his former teammate joined Atletico which was critical of the Catalan club's board.
"The way [Barca] went about it wasn't good and that annoyed Leo, too," Suarez, who left Barca as the club's third top scorer of all time with 198 goals, added.
"He knows what we've been through and how tough it was for us at that moment. I am not surprised he supported me in public because I know him so well. I knew the pain he felt.
"There are things that people don't know, but going to Barcelona training and being sent to train on your own because you're not involved in the 11 vs. 11 game they're playing.
"My wife saw how sad and dejected I was. What I wanted was to laugh and be happy on the pitch again and I had no doubts when the possibility of signing Atletico came up."
However, he scored a penalty on Thursday as Uruguay kicked off their qualifying campaign for the 2022 World Cup with a 2-1 win against Chile.