Tonnies is convinced that the club have found the right man to succeed Keller, and praised sporting director Horst Heldt for identifying the 44-year-old Italian.
"He's got my full backing," he told Bild. "He was looking for a fantastic coach and he got one. It is all down to [Heldt] and he did a great job.
"It was looking increasingly unlikely that [Keller] was going to stay beyond the summer anyway.
"Given the latest matches, which we were no longer able to influence positively, we said that we had to [replace him] now."
Di Matteo will kick off his Schalke spell at home to Hertha Berlin on Saturday.