Neil Lennon, recently given the backing of the Celtic board amid severe fan pressure, watched his side race into a two-goal lead thanks to goals from Tom Rogic and Odsonne Edouard.
But familiar failings soon arrived with Milan equalising through a quick-fire double before finishing off the job in the second half as the hosts took advantage of poor defending.
Celtic, already eliminated from Europe before the match, still sit bottom of their Europa League group with one point from five matches.
For Sutton, it was an another example of the issues which existed all campaign, as he tweeted after the match: “Europe has been an absolute embarrassment this season... we learnt nothing from tonight other than Celtic still can’t defend and the goalkeepers don’t make saves... buck stops with the manager but the players actually need to show some b***s...”