Yet, Mowbray claims his job has been just as hard as he expected when he took charge and resolved to stamp his own identity on the team.
"Change is difficult in any walk of life, in any business, any industry," said the 46-year-old, who is attempting to bring a more attractive style of football to Parkhead.
"With the glare and publicity of the football industry, it is more transparent, I suppose.
"But I felt change had to happen and, as you go along, it's not always a bed of roses and not always win, win, win.
"What I do know is that when you get it right, you will expect to win, win, win."
He added: "The only reason I'm here, I'm assuming, is because of my past record of being able to change teams around and be successful along the way.
"It doesn't always happen straight away."
Victory tomorrow would keep alive Celtic's most realistic hope of silverware, the Active Nation Scottish Cup.
Mowbray takes his side to Irn-Bru First Division side Dunfermline in a fifth-round tie he arguably cannot afford to lose.
Confidence in the camp took another hit after Tuesday night's shock 1-0 defeat at Kilmarnock, a result which left Celtic 10 points adrift of Rangers in the Clydesdale Bank Premier League.
But Mowbray does not believe the visitors will be more jittery as a result tomorrow.
"I don't think there are any nerves," he said.
"If you break down Tuesday night, it was a game very similar to what we've been seeing for the last couple of weeks or so - or probably longer.
"We'd got lots of the ball, lots of chances, lots of possession, lots of positive play and, yet, we didn't quite get the result.
"We go to Dunfermline understanding that we'll probably have slightly more of the ball than they will; we'll probably create more chances than they will.
"But, ultimately, it doesn't guarantee the result.
"So, we've got to work extremely hard and try to make sure that our positive play is rewarded with a victory."
He added: "Morton in the last round was very, very tough. They're in the same league, it's away from home - there are no easy games.
"We've got every respect for Dunfermline.
"They're a good footballing team as well and they've got some good, experienced players, who know what the SPL's all about.
"So we're expecting a tough game and we go there and try to progress in the cup."
Striker Diomansy Kamara, one of eight new faces drafted in by Mowbray during the last transfer window, believes Celtic will be under pressure tomorrow - but no more than usual.
"If you play for Celtic or Rangers, you need to win something and you need to do well every game," he said.
"If you play against small teams or big teams, it doesn't matter; you need to play well always.
"Sunday, we'll have pressure but it's normal for a Celtic player."
On-loan Fulham forward Kamara, 29, admits it has been more than five years since he felt such a weight of expectation at a club.
But the former Modena star is relishing the challenge, saying: "I love this when you play and you have the supporters behind you.
"Because when the teams do well, it's unbelievable.
"Pressure is part of football and, for me, it doesn't make a big difference.
"I'm so happy when I have pressure on my shoulders.
"At Fulham, it was very quiet because they had a small number of supporters.
"But I played in Italy for four or five years and you have a lot of pressure there."