The 26-year-old spent 18 months at Oakwell and was top scorer as the Reds won promotion from League One last season.
“I have had two really good years and it is something I can look back on and say I gave it my all,” said Moore, who has signed a three-year deal at the DW Stadium.
“That form will carry me on to this team and I will replicate what I have been doing for the past two years to here.
“We had a brilliant year at Barnsley last year, to get promoted off the back of a relegation. I will be leaving with a heavy heart, but I feel like this is a new challenge for me and one I’m looking forward to.”
Moore missed Barnsley’s opening day win over Fulham.
A Reds statement read: “The striker expressed a desire to leave Oakwell and after receiving a substantial offer, the club have reluctantly accepted.”
The striker will train with his new team-mates for the first time today.
“It has been in the pipeline for a long time,” added Moore. “So to finally get the deal over the line and sign here is a brilliant feeling.
“I signed because of the way the club is going.
“I’ve had chats with the manager (Paul Cook) and the chairman and everything about it just seems right.
“Since the interest was made I have been really excited about the opportunity and thankfully it’s all come about.
“The gaffer is an infectious character, a really nice guy and I’m really looking forward to working with him.
“The staff have taken me through it all and they have been showing me what they want to see through my style of play.”