The Northern Irishman was appointed as the Bhoys' new boss on Friday.
The 43-year-old was confirmed as Ronny Deila's successor on Friday. And with the Norwegian's tenure coming in for criticism following their inability to do anything of note in Europe, Rodgers will be aware that success on the continent is almost a must if he wishes to be deemed a success.
But Gillespie - who also played for another of the clubs that Rodgers has managed, Liverpool - suggested that the Northern Irishman has the ability to not only take the Hoops into the Champions League, but to also potentially take them into the knockout stages.
"I know how hard it is for Celtic to get through the Champions League qualifiers and get into the group stages. But if anyone can get them through that it is Brendan," he told the Evening Times.
"And I would go so far as to say that if Celtic get into the group stages then Brendan will have a right good crack at getting them out of them. He is a fantastic manager, he gets the best out of players, he knows the game and he will have Celtic playing the kind of football that the fans love."
With the Bhoys only making it to the group stages in one of the past five seasons, it is surely fair to say that their inability to regularly reach that stage has been one of the main reasons why several of their key men have left for Premier League sides.
As Celtic proved this past season that they do not need to be at their best to comfortably clinch the league title - though that they may change in the coming years, depending on how strong Rangers become - competing in the Champions League provides the level of competition that the squad do not come up against domestically, so when they have fallen out of Europe, motivation may well become a factor.
But Gillespie - who was part of the Reds squad that won the European Cup in 1984 - is confident that Rodgers is the man to make them important players on the biggest stage once again.