The 39-year-old has been on the brink of becoming Simon Davey's successor at Oakwell ever since he expressed a desire to join them after they approached him last week.
That led to Rotherham placing Robins on gardening leave until a compensation package could be agreed, something which has now happened allowing Robins to make the short journey across South Yorkshire.
He takes his backroom staff of John Breckin, Steve Taylor and Nick Daws with him.
A Rotherham statement read: "We want to assure our supporters that we are now fully focused on finding a new manager and backroom team who will help us achieve our ambitions and that process will begin immediately."
Former manager Ronnie Moore has already expressed his interest in replacing Robins after losing his job at Tranmere, while former Republic of Ireland boss Steve Staunton also stated his intent on taking the post.