Chants of "Lennon, Lennon, get to f***" were clearly audible as the Celtic manager walked into his post-match media conference as the pressure mounted on the former Hoops skipper following a run of two wins in 10 matches.
Footage on social media showed some fans kicking down temporary barriers that were quickly erected to separate them from those leaving the stadium, and police said two of their officers had been injured in the disturbances.
Sutton also rounded on the anti-Lennon protesters, adding: "The people protesting outside Celtic Park can't call themselves Celtic fans".
The under-fire gaffer, speaking to Celtic TV, admitted: "I'm bitterly disappointed with the result".
Granted time to turn things around in the belief home games with Ross County, St Johnstone and Kilmarnock could spark a recovery, Lennon could even be sacked on Monday after making it clear he won't resign. They are a disgrace.
"Neil Lennon can't stay on after that, but I think he deserves a lot more respect than fans gathering outside the ground". Singing for his head? "Come on. That's not good enough".
None of that will help Lennon's case with the curtain now nearly certain to come down on his second spell at the club, possibly even as early as today. We have lost one trophy, that's gone. It always is every time we go into a new season.We are out of Europe and it's not been great but there is plenty of time to turn it around.
'But it's alright me saying that.
But he still insisted he is the man to lead Celtic to a 12th consecutive domestic trophy when they take on Hearts in last season's delayed Scottish Cup final on December 20 - and that he could reel in Rangers in the title race. It was the most embarrassing defeat for a long time. "It's okay saying, 'We are going to turn a corner".
Asked if he would understand if the Celtic board took the decision to remove him, Lennon said: "No, I think it's too soon for that".