Croatia striker Ivica Olic's second-half goal breathed life into Bayern's fading dream of reaching the knock-out phase alive at Munich's Allianz Arena.
After group leaders Bordeaux enjoyed a 2-0 win over Juventus, Bayern is now just one point behind the Italians in Group A and must win their final game at Turin's Stadio Olimpico on December 8 if they are to progress.
Having drawn their last three league games, the victory was crucial for van Gaal, who has been told he must win the four remaining Bundesliga games before Christmas to keep his job.
"Of course, we are all very relieved to get the win," said van Gaal. "The team played well and we are pleased to have got our destiny back into our own hands."
Van Gaal suspended Bayern's discontented striker Luca Toni on the eve of the game after the World Cup winner told the Italian media the pair's relationship has completely broken down.
Bayern later fined Toni €25,000 for his comments, but goalscorer Olic insisted the squad are behind their coach.
"We are 100 percent behind the coach, everything he does is good and I hope we win everything up until the winter break," said the Croatia star, after making his second appearance after injury. “I think it was very important that we won and put this dark period behind us. I had a bad injury and it was a long break and it was pleasing to get the winning goal.”
Glum-faced Toni spent the game in the stands next to French midfielder Franck Ribery while Bayern's fellow walking-wounded Arjen Robben and Miroslav Klose looked on.
But with Germany's rising stars Thomas Müller and Mario Gomez causing the Israeli team constant problems, in the end Bayern's superior fitness made the difference.
Bayern took the lead in the 62nd minute when captain Mark van Bommel found Gomez in the penalty area and he fired a shot to the left of Haifa goalkeeper Nir Davidovitch.
The Israel national goalkeeper made the save, but the rebound went to Olic, who slotted home to keep the dream alive.
"We are of course very pleased to be back in business and everything is back in our own hands," said 20-year-old Müller. “We needed a lot of passion and fire, it was noticeable that everyone is concentrating hard."