During the summer, Barca missed out on Coutinho as Liverpool opted to keep a player who had only just penned a new long-term contract, but the Premier League side have now allowed the Brazilian international to sign for the Catalan giants for a reported £142m.
The transfer could have been potentially dragged out until the summer - a year after Barcelona's initial interest - but Bartomeu has said that Coutinho's mindset was key to completing the move at the start of 2018.
At a press conference, Bartomeu said: "All the Barca fans in the world are happy today to meet Philippe Coutinho. We've tried to get him to come and play for us since last summer.
"He was patient and that was the determining factor in bringing him here. I'm very happy to announce the signing of one of the stars of world football, we hope for a long time.
"Everyone knows him as a player. But what I can say is that it wasn't easy to bring him here. But he wanted to be here and he's part of the new project with Ernesto Valverde. He has a lot of family with him here today which everyone wanted and that's what we've done to make it happen."
Coutinho will not make his debut for Barcelona for a number of weeks after sustaining a thigh injury.