The 39-year old Italian has signed a 12-month contract with the Baggies after just one season at the helm at MK Dons.
Despite being disappointed to lose Di Matteo, MK Dons were aware his qualities would be sought after at a higher level.
"Following a formal request to speak to Roberto on Monday, we reluctantly allowed Albion officials to approach him with regard to the position," a statement from MK Dons read.
"Having enjoyed an extremely successful first season in Milton Keynes, his first managerial position, it came as little surprise that Robby's undoubted qualities would prove attractive to a club at a higher level.
"While we are extremely disappointed to lose our manager so close to the pre-season, we recognise that the opportunity at the Hawthorns represents a big step forward in Robby's career.
"We will now begin the process of appointing a new manager that can build on the success of recent seasons and continue to take the club forward."