Bayern, whose European performances in early season have contrasted with some inconsistent displays in the Bundesliga, were missing half a dozen first choice players through injury.
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"We have not had much luck so far this season but tonight we were very lucky," said Bayern coach Louis van Gaal.
"But it is also hard when the opponents play with 10 men at the back. Overall I think we deserved to win," he told reporters.
LIVE:Minute-by-minute coverage as it happened
Cluj, third in the group on three points, were bolder at the start and took a deserved lead when Portuguese defender and captain Cadu headed in a Roberto De Zerbi free kick at the far post in the 27th minute.
Felice Piccolo had missed from almost the same spot minutes earlier, exposing Bayern's makeshift defence.
With injured captain Mark van Bommel, Ivica Olic, Miroslav Klose, Arjen Robben and Franck Ribery watching from the stands, the hosts got an unexpected equaliser on 32 minutes when Cadu deflected a Toni Kroos effort into his own goal.
Worse was to follow for the Romanians as five minutes later defender Cristian Panin chested in a Bayern corner kick for their second own goal.
The Romanians refused, however, to give up and came agonisingly close to an equaliser just before the break when Lacina Traore beat the offside trap to race clear and rattle the post with a low drive.
Bayern striker Mario Gomez notched his fourth goal in two games, after scoring a hat-trick at the weekend, to make it 3-1 in the 77th minute.
Culio cut the deficit with four minutes remaining, beating keeper Hans-Joerg Butt with a well-struck shot but Cluj failed to grab what would have been a deserved equaliser despite some desperate last-gasp attacks.
"We have to be thankful today because all our goals were a bit strange," said Gomez, who has spent much of the season on the bench. "But football is like that. Sometimes it does not work for weeks and then suddenly your luck is back."
