But he admits the new Hoops boss has a tough act to follow, with his predecessor Gordon Strachan delivering three championships in four years in the east end of Glasgow.
Former West Brom boss Mowbray was unveiled to the media on Wednesday and received a rapturous reception from fans who had gathered outside Parkhead to welcome their new manager.
"I was delighted that Tony got the job," said McCluskey, who played for the Hoops between 1975 and 1983.
"He is one of the most imaginative coaches. The fans were delighted today and he got a great reception at Celtic Park.
"I'm sure he has given everybody at the football club a lift. The reception he got was amazing.
"It's a big, big lift for the football club after being down for the last few weeks."
Celtic - and Strachan - narrowly missed out on a fourth successive title when Rangers triumphed on the final day of the season.
Runners-up spot means Celtic face the prospect of qualification in a bid to join their Glasgow rivals in the group stages of the Champions League.
With the first qualifier set to take place at the end of July, there is little time for Mowbray to adapt to his new surroundings before being pitched in at the deep end.
"It's going to be very difficult for him," said McCluskey.
"He's only got six weeks until the first qualifying game in Europe.
"There is a lot of hard work ahead for him and Gordon Strachan will be a hard act to follow after winning three league titles.
"But I'm sure Tony is looking forward to it."
Mowbray earned a reputation for attractive, attack-minded football at both Hibernian and West Brom and McCluskey believes Celtic fans can look forward to that philosophy being implemented at Parkhead.
"He has the mind-set that he likes to play attacking football and I'm sure that won't change at Celtic," said the former striker.
"It's what Celtic fans have been brought up on so it's good news for everybody all round."
And he believes Mowbray's Celtic background - he spent four years there as a player in the 1990s - will do him no harm either.
"I don't think it's vitally important but it helps, it definitely helps," said McCluskey.
"It helps him to know what he's going into, the size of the club.
"Peter Grant is also coming back so it will be great."