The 46-year-old has swapped a promotion push with Rovers for a relegation battle with the Owls after becoming the club’s sixth manager since the summer of 2015.
Wednesday are six points adrift of Championship safety with 14 games left in the season, with Moore’s first match in charge being Wednesday’s derby relegation six-pointer with Rotherham United at Hillsborough.
In a statement, Rovers chairman David Blunt expressed the club’s disappointment at the developments - at the timing at such a critical stage of the season.
Meanwhile, the club have named Andy Butler as their new manager with immediate effect.
Wednesday made an approach to Rovers to speak to Moore on Friday, with Moore's representatives informing the club on Saturday morning that he wished to speak to the Owls - ahead of making a decision to take the job on Sunday night.
Rovers have received a six-figure compensation fee for Moore.Moore's assistant Jamie Smith has also left the club with immediate effect.
Moore said: “I understand David’s comments in terms of being disappointed at the timing. But I am sure he’s delighted for me as an individual.
“From when I joined Doncaster, first and foremost, I want to say what a wonderful football club it is and it was an honour to go back and manage as a former player.
“I leave the football club in great hands and I know Andy Butler has taken the reins now for the rest of the season and I am really pleased for Buts as it’s the continuation of the work we were doing. They are in good hands and the club have an exciting finish to the season to look forward to.”
Meanwhile, Moore insists that the lure of taking over at Hillsborough proved impossible to resist and he says he had no hesitation in joining the club, despite their parlous position in the bottom three.
Moore added: “The chairman spotted and identified me as one of the candidates to take the club and try and lead it forward and when the opportunity came, I didn’t hesitate.
“It is an excellent club steeped in huge tradition with the size and structure of the club. The opportunity was too much to resist and I had little hesitation in wanting to become the next manager.
“There is not much time to think and we must hit the ground running with a game on Wednesday.
Urging everyone connected with the club to unify and come together amid a difficult time in the Owls history, he continued: “It is a huge, huge challenge. But what has gone on has gone now.
“But now we can come together as one as we all have a desire and love for this wonderful football club.
“The word I use unity and a oneness for us all.”