Gennaro Gattuso was set to have serious questions asked of his tenure until own goals from Stelvio and Tom Schnell either side of Hakan Calhanoglu's measured finish in the 70th minute rescued a second-half deficit.
Fabio Borini came off the bench to a polish to a scoreline that was highly unflattering when Dominik Stolz and David Turpel struck in the space of 10 minutes to give Dudelange the lead following Patrick Cutrone's opener.
Though undoubtedly far from happy with their performance, the Rossoneri can comfort themselves with knowing they only need a point away to Olympiacos on the final matchday to join Real Betis in advancing from Group F.
The host began with admirable intent and it was no surprise to see Cutrone squirm a 21st-minute shot through Landry Bonnefoi following Gonzalo Higuain's clever backheel.
But the early goal only seemed to lull Milan into a slumber that was punished six minutes before the break, Schnell making a nuisance of himself from a corner and midfielder Stolz emphatically dispatching the loose ball.
Gattuso must have been ready to explode when, four minutes after the interval, Cristian Zapata misread a deflection and left Turpel to unerringly pick out the bottom-left corner.
Then came the much-needed response. Cutrone restored parity thanks to a heavy touch off Stelvio in the 66th minute before Calhanoglu stroked home from 20 yards moments later to complete the quickfire turnaround.
The relief was palpable when skipper Schnell headed into his own net at the near post as Borini was left to finalise a result that keeps Milan one point off leaders Betis.
What does it mean? Milan poised to progress
There is only so much the final result will do to gloss over a largely lacklustre performance, but Gattuso will at least be thankful to occupy a strong position leading into the decisive clash with Olympiacos, whom his side beat 3-1 in the reverse encounter.
Cutrone instigates the comeback
His fourth goal in Europe this term gave Milan a confident start and it was young striker Cutrone who stood up when circumstances were more difficult, posing constant problems amid the second-half carnage.
Bonnefoi buckles under pressure
Dudelange needed everything to go its way in this match and it might have managed a huge upset if not for a few questionable goalkeeping moments.
Bonnefoi's failure to grasp a Cutrone shot that was well within his range set the visitors on the back foot and he did not cover himself in glory with either of the next two goals that helped deny Dudelange its dream.
Milan hosts Parma and Torino before heading to Greece for its final Europa League fixture on December 13, the same day that Dudelange rounds out its campaign at home to Betis.