The 39-year-old was placed on gardening leave by the Millers last week after expressing his desire to speak to the Championship club.
And the move has now been completed after the two clubs settled on a figure to secure Robins' release from United.
The former Manchester United striker, who spent two-and-a-half years in charge of Rotherham, replaces Simon Davey at Oakwell following his departure on August 29.
Robins takes his backroom staff of John Breckin, Steve Taylor and Nick Daws with him.
Robins told the club's official website: "I am delighted to be here after a protracted few days and I can now start the job.
"It has been frustrating and I have been kept waiting in the wings a little but at the same time I am thankful it has all been sorted out and resolved.
"I have my own coaching staff and we are looking forward to getting started and seeing what the players can do.
"We have to get points on the board quickly. We are playing catch up and our season starts now."