Gordon joins the reigning La Liga champions for a fee of €80m (£69.3m) that makes him the second most expensive player Newcastle have sold in their history.
The England international joined the Magpies from Everton in January 2023 and made 152 senior appearances for the club, scoring 39 goals.
He was Newcastle's leading goalscorer last season, scoring 17 goals in all competitions, and is set to play alongside Lamine Yamal and Raphinha in Barcelona's frontline next term.
"I owe this club a lot because, when I arrived, I was quite lost both in life and football. The club has given me a sense of belonging and a sense of identity. It's allowed me to do what I always thought I could do. It's put me on the biggest stage and allowed me to perform for the shirt.
"Since coming to the club, I feel like I've improved a lot on the pitch but this club has played a big part in the person I've become over the last three-and-a-half years.
"It was really important for me to leave this place in a good way because I've loved every single minute of being a part of Newcastle United. This is an incredible club and one that I'll never forget. I'll be a fan for the rest of my life."
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It became increasingly clear that Gordon had his heart set on leaving Newcastle in the second half of the season, hence Eddie Howe's decision to leave him on the substitutes' bench for the final four games of the campaign.
It had initially looked like Bayern Munich were leading the race for his signature, but with the German club refusing to meet Newcastle's demands, Barcelona swooped in to seal a quickfire deal.
"While we're disappointed to lose Anthony from our squad, we understand that this is a big opportunity for him," said Howe.
"He has been a big part of our success in recent years, and the fact that he developed into an England international during his time at Newcastle United is a testament to the hard work he invested in his development with the support from everyone at the club.
"He leaves with our best wishes, and I am confident that he will go on to be a success, both with Barcelona and the national team at this year's World Cup."
Gordon's departure is the first big transfer move during Ross Wilson's tenure as sporting director. Wilson will not be tasked with investing the money received to help strengthen Newcastle's first-team group this summer.
"Anthony has been a fantastic professional during his time at Newcastle United," said Wilson. "The relationship between him and the club, particularly with Eddie and our players and staff, has been excellent throughout.
"Anthony has shown real growth and development in his game. That is a reflection of both the environment we strive to create for our players and, importantly, Anthony's own commitment to improving every day.
"There's been interest in Anthony from a number of Europe's leading clubs over a period of time, and the conversations involving the club, led by David Hopkinson and myself, as well as Anthony's representatives, have been conducted confidentially and with a great deal of professionalism on all sides.
"I've enjoyed getting to know Anthony and enjoyed our relationship together. We thank him for his contribution and wish him and his family all the very best for the future. They will always be welcome back at St James' Park."