Former Celtic striker McGhee had hoped to take over at Parkhead himself before rejoining one of his other former clubs and has been granted an immediate chance to prove that he would have been the better choice.
Rangers host Celtic in the season’s first Old Firm clash on 3 October, with the traditional New Year fixture at Parkhead on 2 January, while Ibrox hosts for the third derby of the season on 27 February.
Mowbray will face his former club Hibernian, now under the charge of former Falkirk boss John Hughes, for the first time on 29 August at Easter Road.
Celtic are at home to the Dons on 5 December and away at Pittodrie again on 13 February.
Hearts visit Celtic Park on 19 September and 10 February, with the return fixture on 19 December.
Millwall manager Neil Harris was delighted with his side as they comfortably won 2-0 at Barnsley.
A three-goal burst inside eight first-half minutes ensured Lincoln secured derby bragging rights with a 3-1 victory over struggling Grimsby.
Millwall's unbeaten run of games stretched to eight in the Sky Bet Championship as they comfortably defeated Barnsley 2-0 at Oakwell.