United rattled the woodwork twice, but could not find the net and had to settle for their seventh goalless draw of the season.
Chelsea had their chances too. David de Gea pulled off stunning saves to deny Pedro, Cesar Azpilicueta and John Terry - and Nemanja Matic ploughed the ball into the East Stand after going through after the break.
But, United were the better side and probably deserved three points.
Whether that is enough to save Van Gaal remains to be seen. The statistics do not look good for the Dutchman.
United are now winless in eight, they go into the new year in their lowest position since the 1989-90 season and have scored two goals in their last four.
Some fans made it clear they wanted Jose Mourinho, rather than Van Gaal in the home dugout. One held a banner aloft imploring the manager to "Leave, Vanish, Go" and red Mourinho scarves were on sale from unofficial vendors outside the ground.
Van Gaal's self-doubt was clear to see after the Stoke defeat, but on Monday he strode down the touchline with his chest puffed out and head held high while waving to fans.
Whatever Van Gaal had said to his players before kick-off worked as their first-half performance bore no resemblance to the turgid, predictable displays so often seen here this season.
Just three minutes in, Wayne Rooney, restored to the starting XI, laid the ball off to Juan Mata, who hit the bar with a fierce volley.
The United skipper then provided Morgan Schneiderlin with a chance to score from the edge of the box, but his half-volley flew just wide.
This was thrilling stuff from the hosts, but Chelsea had their moments too. Willian swung over a corner which Terry met with a diving header. Luckily for United, De Gea saved athletically.
United continued to attack the Blues and had most luck down the flanks. Ashley Young and Matteo Darmian overlapped to cause Branislav Ivanovic and Azpilicueta problems. In the 12th minute, Darmian raced to the byline and sent over a dangerous cross, but there was no one there to convert.
Ivanovic was then turned inside out by Anthony Martial, who smacked a low shot on to the near post.
John Obi Mikel had to resort to illegal means to stop Martial. The midfielder stuck out a leg and halted the forward's pacy run just before the half hour.
Rooney was full of confidence. He took a touch before unleashing a fizzing drive which Thibaut Courtois did well to tip over.
The home supporters were enjoying this entertaining spectacle, but their team could not find an opener. Just after the break, they were biting their nails as Chelsea poured forward. De Gea saved brilliantly from Pedro and Azpilicueta in quick succession.
Then it was Courtois' turn to keep his team in it. The Belgian dived to his left to deny Herrera from point-blank range. Willian got away with a handball in the box as Chelsea cleared.
Matic was then presented with the best chance of the match when he raced through, but he skied over.
Rooney could have snatched the game at the death but he volleyed over. There was a smattering of boos after the final whistle, but no widespread dissatisfaction despite an eighth game without victory.
Tweet of the match:
"Is it me or is this Man U v Chelsea game average ? to say the least." - It's not just you, @reid6peter.
David De Gea: 9 (Out of 10)
Matteo Darmian: 7
Chris Smalling: 6
Daley Blind: 7
Ashley Young: 7
Morgan Schneiderlin: 7
Bastian Schweinsteiger: 7
Juan Mata: 8
Ander Herrera: 7
Anthony Martial: 8
Wayne Rooney: 8
Cameron Borthwick-Jackson: 6
Memphis Depay: 6
Phil Jones: 6
Thibaut Courtois: 8
Branislav Ivanovic: 5
Kurt Zouma: 7
John Terry: 7
Cesar Azpilicueta: 5
Nemanja Matic: 5
John Obi Mikel: 5
Eden Hazard: 5
Ruben Loftus-Cheek: 5
David de Gea: Saved superbly to Deny John Terry's rapid header in the first half and then after the break saved twice in quick succession from Pedro and Cesar Azpilicueta.
MOMENT OF THE MATCH:
Wayne Rooney's miss: Rooney justified his return to the starting XI with an improved display. The striker exuded confidence and tested Thibaut Courtois with a 30-yard piledriver, but in the final few minutes, he should have claimed victory. Cameron Borthwick-Jackson swung over a deep cross and there was Rooney, unmarked a few yards out, but he shot wide.
VIEW FROM THE BENCH:
Louis van Gaal did the right thing in recalling Rooney and his decision to bring Morgan Schneiderelin back into the team was a wise one. The Frenchman helped turn defence into attack quicker and Anthony Martial looked threatening cutting in from the left. Without a fit striker, Guus Hiddink played Eden Hazard up front. The Belgian struggled throughout, but was starved of service. Kurt Zouma gave Hiddink food for thought with an assured display at centre-half.
MOAN OF THE MATCH:
Why have United waited until now to start playing the attacking brand of football we saw in the first half? The opening 45 minutes was by far the best United fans have seen this season. Even though United did not score, they had pace and guile down the flanks, as well as flair and penetration down the middled. Why has it taken until possibly Van Gaal's last game for the shackles to come off?
WHO'S UP NEXT?
Manchester United v Swansea (Barclays Premier League, Saturday, January 2)
Crystal Palace v Chelsea (Barclays Premier League, Sunday, January, 3)
Van Gaal had hinted after the Stoke game that he could decide his own managerial fate, but suggested after this performance he would not walk away.
Asked about his future, the under-pressure Dutchman told BT Sport 1: "It's for me not an interesting subject because I cannot have any influence on that.
"I can only work with my players and you have seen that the players are fighting for me; they have given a good performance.
"Even after this result the fans were applauding. I cannot say that I am very much concerned, but in this football world you never know. I have full confidence in the board and that's also more important I think in my players.
"When the players can give such a performance with this lot of pressure then it is not any reason to resign for me. Maybe the media wants that, but I shall not do that."
Pushed on whether he would definitely remain, he responded: "I am under contract so it's not a question of staying, it is a question of fulfilling my contract."
United skipper Rooney admitted the players owed it to themselves to produce an improved performance .
"We'd be letting ourselves down if we didn't go on to that pitch and let ourselves play football and give our all," the England striker said.
"You've seen the fans today - they follow us everywhere. We wouldn't be professional players if we didn't do that. We have to fight to the end.
"We're in a tough time, everyone knows that. We'll keep fighting and we'll come back again, we know that. But at the minute it's tough. We have to just keep working hard and I'm sure the victories will come."
Chelsea were missing Diego Costa to suspension and a number of injured players, so Hiddink was happy to emerge with a share of the spoils.
"I'm happy with a point because they were difficult circumstances for us, we had a lack of striking people - they are all injured," he said.
"I can be happy with the performance of the team. We would have liked to score. In general, due to the circumstances, I'm happy with the performance."
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