Dons manager Derek McInnes and assistant Tony Docherty opted to stay with Aberdeen despite talks this week with Championship club Sunderland.
"I was fairly confident he would stay," Milne told BBC Scotland on Friday.
"He phoned yesterday morning and said he was absolutely committed to the club. It would have been a massive wrench for him to walk away."
McInnes, 45, took over from Craig Brown in 2013 and the following year led the club to League Cup success, beating Inverness Caledonian Thistle in the final for the Dons' first trophy in 19 years.
Last season, McInnes and Docherty guided the Dons to second place in the Premiership behind Celtic, and were runners-up to Celtic in the Scottish Cup and League Cup.
"He sees Aberdeen Football Club as his," said Milne of McInnes who, along with Docherty, is contracted to the Pittodrie club until 2019.
"I've built up a very close relationship with Derek over the last three or four years and we have been in constant dialogue over the past four or five days. I believed it was 50-50 (whether he would leave).
"He has always said how happy he is at Aberdeen. He feels he gets the backing of the board, he has been very successful here, any move entails risk and he knows what he's got here.
"Last year was probably the best season in the last three decades and he believes that he can better last year.
"I'm absolutely delighted. I've always said we had one of the best management teams in the whole of the UK here."
Milne revealed that the club had "reshaped" McInnes's contract in the "past week or so" and that fans can expect to see a few new players coming to Aberdeen when McInnes returns to work on Tuesday.
Milne also says he would be surprised if the club don't make a few signings before the end of next week.
Aberdeen chairman Stewart Milne thinks the club have retained "one of the best management teams in the UK".