Croatian Ivica Olic tapped in the winner on the rebound in the 62nd minute after Mario Gomez had his shot palmed away by keeper Nir Davidovitch.
Bayern, who dominated possession throughout and squandered several good chances, now need to win their final Group A match away against Juventus to secure qualification.
Haifa, bottom of the group, have yet to score a point or a goal in the competition.
"I saw a very good match against a very tough team," Bayern coach Louis van Gaal told reporters. "It could have been more than just one goal. I think it was one of our best games.
"I think now the chances are 50-50 against Juventus," he said. "We will get our chances in Turin."
The Germans were without injured playmaker Franck Ribery and Dutchman Arjen Robben as well as Miroslav Klose, who injured his elbow on Tuesday.
Italian striker Luca Toni was surprisingly left out of the squad after airing his dissatisfaction with van Gaal in the Italian media but the Germans were still the livelier side and twice came close through Gomez.
CLEAR CHANCES
The Germany international passed up a clear chance to score in the 17th minute when he dropped his marker and connected in the box with a Mark van Bommel cross only to see his shot fly wide.
Gomez tried his luck from 18 metres but his low drive was also slightly off target.
Bayern, who had not scored a goal in their two previous home games in the Champions League, had over 70 percent possession for the first hour but only managed to hit the bar through Bastian Schweinsteiger.
It took them 62 minutes to break the spirited Israeli defence when captain van Bommel found Gomez at the edge of the box.
His shot was saved by Davidovitch but Olic, back only days ago from a seven-week injury break, lurked at the far post to tap in.
