Ange Postecoglou insisted he was "as surprised as anyone" when Dominic McKay departed as Celtic chief executive on Friday.
McKay had only officially taken over from Peter Lawwell on July 1 but the Parkhead club cited "personal reasons" for him stepping down, with Celtic's Director of Legal and Football Affairs, Michael Nicholson, appointed as acting chief executive officer.
Postecoglou, who sat alongside McKay on the day he was confirmed manager in the summer, was speaking after the 3-0 cinch Premiership win over Ross County at Parkhead, which came courtesy of a deflected goal from debutant defender Cameron Carter-Vickers and a brace from striker Albian Ajeti.
The Celtic boss confirmed left-back Greg Taylor had gone off in the second half with a recurrence of a shoulder injury which makes him a doubt for the Europa League tie against Real Betis on Thursday.
"We have to be careful with the narrative of who brought me to the club," said Postecoglou, when that premise was put to him.
"I was surprised as anyone that Dom resigned but I am going to be respectful of him and his family.
"When people cite personal reasons, there's reasons for that. Michael is now in charge and that's good for me because I already know him.
"We are all transient figures, we all pass through at this great club. The constant is the fans and they expect us to get on with it. I've done that throughout my career."
On Taylor's injury on the day he returned, the manager said: "He had two weeks of rehab.
"It's a bit of a concern for him that he's re-injured it but we'll leave that to the medical people."
Postecoglou added: "The result was positive. The performance was OK, we started really well and just needed a goal.
"In these games, if you don't score early it gives encouragement to the opposition and they hung in there and defended well at times.
"I've got to keep things in perspective. We're giving guys debuts and it was Albian's first start so we're nowhere near as fluent as we can be."
Ross County boss Malky Mackay said the afternoon was one of "slight frustration".
He said: "You are coming to play against a top team with some fantastic players in front of 50,000 people but we had a game plan and stuck to it.
"We frustrated them in the first half and I thought we countered well, but when you counter you have to be clinical and you have to score when you have the chances here.
"I was obviously really disappointed with the deflection, which you can't legislate for, and at that point Celtic were looking a little bit frustrated and the crowd were frustrated.
"When that goes in it relaxes them, and at the same time I thought we broke really well at that point and we had a glaring chance when Joe Hart makes the point blank save from Dom (Samuel). These are the fine margins you have to execute here."
