Aston Villa manager O'Neill, who was at the helm at Celtic between 2000 and 2005, also insisted the 38-year-old Lennon got the job on his own merits and not because he or anyone else had sung his praises to the Parkhead board.
He said: "Listen, Neil has got that job because of what he achieved in the short time he was given the post in the interim, as simple as that.
"It is nothing to do with anything else or anyone else. People can make up their own minds and they did that through the evidence that was put in front of them.
"Winning eight consecutive league games is no easy feat, no matter who you are or what league you are playing in.
"If those games hadn't gone so well it wouldn't matter a jot what anyone had to say to the world about Neil. He has got there on his steam, no question about that.
"He deserves to be in the position he is in because he would have had his eyes opened in those nine games he was in charge.
"People may talk of a lack of managerial experience, but he has been there and experienced just about every conceivable emotion as a Celtic player – incredible highs and incredible lows.
"That will stand him in good stead and, for me, he is the man to bring a bit of success back to the club. I honestly believe he will be a hit as Celtic manager."
O'Neill also dismissed speculation that he was ever in the running for a return to Parkhead.
The Villa manager rubbished the rumours that he had been set to come back either as manager or as mentor to Lennon.
"No, not at all, there was never any suggestion of anything like that," he said yesterday when asked if there had ever been any likelihood of him returning to Parkhead.
"I loved my time at Celtic, I really did. It is a special club with special supporters and I have a lot of very fond memories of the years I spent there. I worked with some great people, had good relationships with many of the people at the club, and in particular, (owner) Dermot Desmond, and I had a strong friendship.
"I continue to hold the club in the highest regard, but this is now all about Neil. It is his time and he deserves to be in the position he is in.
"The club are to be congratulated for giving him the job. Anything is a risk in football and, so far as I am concerned, he is absolutely the right man to take the club forward now. Like everything, he will need a little bit of luck to go with him, but when it comes to someone going in there and lifting Celtic again, Neil is as good as anyone."