Defeat to Hearts at the weekend left the Hoops 10 points adrift of Old Firm rivals Rangers and sitting in third spot in the table behind second-placed Motherwell.
But, having played alongside Neil Lennon during their time together at Parkhead, Sutton is confident the Celtic boss can haul his side back into the title race.
Speaking at the launch of his autobiography, he said: "By Neil's own admission, it was a disappointing performance on Sunday. The team didn't play particularly well.
"They have lost three games this season and only lost four in the whole of last season.
"We are all aware that things need to change pretty quickly but there is no reason why Celtic can't go on a run of games and start pushing their way up the table.
"I went to the game on Sunday and the fans are totally behind Neil.
"Neil talked about character and that's important. It's also about having the ability when they are playing to work out how to win games and that comes with experience.
"During my time at Celtic, we were lucky to have someone like Henrik Larsson, who bailed us out most weeks. They could certainly do with a player like him now but that's not the case.
"Neil's been dealt the hand that he's got. There was a lot more money spent during my time under Martin O'Neill.
"He's in the position he's in and Neil and the players will want to get out of it as quickly as possible."
Sutton believes Rangers have already demonstrated this season just how quickly fortunes can change for the Glasgow giants.
The Scottish champions crashed out of both the Champions League and the Europa League at the qualifying stage but remain undefeated in league action at the summit of the SPL.
The former striker said: "The problem in Glasgow is that after a couple of bad games the heat is on but that's the nature of what it's like here. Things can change.
"A couple of months ago when Rangers went out of Europe, it was all doom and gloom over there.
"A few good results in the league and it's all rosy over there, albeit they got knocked out of the League Cup."
Sutton also believes injuries to key players - such as skipper Scott Brown - has had an impact on Celtic's results.
He said: "He's a big loss. I know he's not everybody's cup of tea and favourite player.
"But he does have the strength of character and aggression to get other people going around him. Certainly, missing a captain like him is a big loss.
"But it's always been the case that the squad is important and players need to step up and be counted."