Griffiths notched the fourth goal after Anthony Stokes helped himself to a hat-trick, the third coming from the spot early in the second-half.
Celtic keeper Fraser Forster saved an Aaron Doran penalty almost immediately afterwards but goals from the former Wolves player, Efe Ambrose and substitute Teemu Pukki, on for Stokes, completed the rout.
Griffiths is under investigation by the police, the Scottish Football Association and Celtic for allegedly joining Hibernian fans, of which he is one, in singing a song calling former Hearts player Rudi Skacel "a refugee" in a pub before the Edinburgh derby last month.
And before the game started, a series of banners were held up by some Celtic fans which read: 'A man must be a celt on and off the field otherwise he is of no value to this club'. Another banner read: 'refugees welcome, created by immigrants' which alludes to Celtic's formation in 1888.
"I can't comment on the banner because I wasn't sure who it was aimed at for a start," said the Hoops boss.
"If it was aimed at Leigh then we are going through a process at the minute and we will be able to speak more about it when the process finishes.
"We've all made mistakes. But I think the kid will come good. I think he'll learn from it and have a good career here.
"I don't know if he was aware that the banner was aimed at him. If he did then then we expect those kind of performances from him anyway.
"I think that's seven goals in 11 games so he's doing exactly what we brought him in to do."
Lennon was delighted with the biggest Scottish Premiership win of the season.
The former Celtic skipper said: "That was up there with our best performances of the season along with Hearts away in the Scottish Cup and Motherwell at Fir Park.
"The attitude and appetite of the players was fantastic. The quality of the football, the pace, the movement - everything was there to see.
"Even my goalkeeper made a brilliant save from a penalty so they were all at it today. So I'm really pleased.
"For a long time now Anthony has been playing very well. Now he's getting the goals that his all-round play is deserving.
"That's his 19th of the season - he just needs one more for 20.
"More or less every season he's been here he's reached 20 goals so he's been a very important player since I came in the door and he's improving all the time."
Stokes has scored eight goals in his last six games and with three matches remaining is confident of reaching the 20-mark at least.
The Republic of Ireland striker said: "My form has been solid the last few weeks.
"We've three games left and I just go out and try and do my best and try to get over 20 goals now.
"I felt I was playing well in games but the ball just wasn't breaking to me at times, I went on a bad run of maybe 10, 12 games possibly more, where I didn't score but I felt the performances were there.
"At this minute it's just confidence as well. As a striker I think that plays a big part and if the ball breaks to you if you're in good form it tends to hit the back of the net."
A "bitterly disappointed" Caley boss John Hughes insisted that his side are not a team who should be losing 6-0.
He said: "We worked hard on how we would set up but then lost the first goal from a set-piece.
"But you also have to give credit where it's due. Celtic are miles ahead of anyone else in Scottish football and have an abundance of attacking talent. Neil has done a tremendous job here and you have to hope you catch them on an off day.
"In saying that, we also self-inflicted. I felt that when the third goal went in we started to feel a bit sorry for ourselves and lost our shape a bit.
"That's not us. It is disappointing but we have to try and learn from it and respond in the right way."