The Parkhead side's 2-0 victory in the Clydesdale Bank Premier League was marred by trouble in the away end as fans let off bangers and fought among themselves during the first half, with Tayside Police speaking afterwards of a "level of disturbance primarily within the Celtic section of the stadium".
They also confirmed that five supporters, one fan of the home side and four following Celtic, will be reported to the procurator fiscal for "alleged football-related offences".
Mr Lawwell said last night: "We are very proud that Celtic supporters enjoy such a great reputation and it is therefore extremely disappointing that the behaviour of a small minority has tarnished the good name of the club and our fans in this way.
"Clearly, we apologise to Dundee Football Club – we will work with them and look into any incident which is raised and if we can identify any individuals involved, we will obviously take the strongest possible action.
"We have received a number of complaints today from our own supporters about the conduct of this minority. It is time for all our true supporters who care about Celtic to unite with the club and find a solution to this problem."
Jim Thomson, Dundee FC operations manager, said he and his staff had been shocked by the behaviour of an element of the visiting fans, and laid some of the blame with the scheduling of the match, which kicked off at 7.30pm at the behest of broadcasters ESPN.
The fixture was originally envisaged as that of the festive Celtic and Rangers match, before Dundee took the place of 'Club 12' on the fixture list.
Mr Thomson said: "There was widespread drunkenness within the Celtic support.
"We were confiscating drink from supporters in the concourse leading to the turnstiles.
"All in all it was just not a nice experience. In the 20 years I've been organising games here at Dens Park I have never seen anything like that – it was a throwback to the 70s that we are all probably ashamed of. There was almost a riot.
"The game should never have kicked off at 7.30pm on Boxing Day - it was a disaster waiting to happen."