The transfer was in doubt for a spell and the 20-year-old refused to train with Peter Bosz's side in an attempt to force through the move - an act which led to him being suspended by BVB and cost him a place in the most recent France squad, but the switch to Camp Nou eventually materialised.
Mbappe – who himself was the subject of fervent transfer speculation for months before joining Paris Saint-Germain on deadline day in a loan with an option to buy, reportedly for €180m – helped Dembele through the final weeks of the window.
"He's a really good colleague," Dembele told Mundo Deportivo of Mbappe.
"We speak a lot on the phone and even more during this process. He's happy for me.
"He tried to calm me down when the days were passing and the signing wasn't made official. He told me it was a question of time.
"I was nervous about the deal, although I could sleep at night. I just wanted it to go through as quickly as possible.
"He was really happy for me when I signed, he congratulated me, wished me the best. I wish him the best, too."
Dembele was bought as Barca sought a replacement for Neymar following the Brazilian's world-record €222m move to Paris Saint-Germain.
"Neymar is a great player with a lot of class, one of the best in the world," said Dembele.
"I am young, I am here to learn and to add what I can to Barcelona.
"I'm focused on getting up to speed because I hadn't trained for two weeks. Now it's time to get back in shape, which is not too hard given my age and my weight. I don't know if it will be difficult for me to adapt. I hope not.
"They play differently here. It's another way of understanding football. Here it's about possession, passing. I've followed it a lot on TV. I hope to learn to play like this, although my qualities are speed, attacking and one-on-ones."