The 51-year-old and his assistant Brian Kerr have signed contracts until 2029.
Peter Leven had been caretaker boss at Aberdeen following the sacking of Jimmy Thelin in early January.
The Scottish Cup holders lost 3-0 to second-tier Dunfermline Athletic in Saturday's quarter-final and have won just once in eight outings in all competitions.
"There is a strong desire to re-establish a clear identity for our club, to become harder to play against and restore competitiveness in our matches," said chairman Dave Cormack.
"For those reasons we believe Stephen is the right head coach to deliver exactly that."
Robinson said he could not turn down the opportunity.
"Aberdeen has a tremendous history and there's no mistaking it is a huge club," he said.
"The facilities and resources that will be at my disposal are excellent and, more importantly, the fan base is tremendous so I'm excited to work for a club of this size and strive to take it forward."