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.
United fans hold Jose Mourinho scarves at Old Trafford
Manchester United boss Louis van Gaal may still be in his post, but a few Red Devils fans at Old Trafford are already preparing for the arrival of Jose Mourinho, it seems.
Pressure is growing on Van Gaal after his United team endured a dismal run of four successive defeats for the first time since 1961.
If the Red Devils lose to Chelsea on Monday evening, it will be the first time the club have lost five successive games in a single season for 79 years.
Mourinho has been linked with Van Gaal's position following his dismissal by the Blues, and it appears some United fans are readying themselves for the arrival of the self-styled 'Special One'.
A number of United supporters snapped up Mourinho scarves that were on sale from unofficial vendors behind the East Stand prior to kick-off on Monday.
The red scarves, which had Mourinho's face on them, were clearly not official club merchandise, but a few home fans still bought them prior to kick-off.
The scarf-seller, who was asking £10 for the scarves, yelled "Welcome to Old Trafford, Mourinho!"
Van Gaal, who hinted he could resign after the Boxing Day defeat to Stoke, pulled no punches in his programme notes, writing: "There is no disguising the fact that we are in a very bad period, the worst I have known since I became Manchester United manager, and I know things must improve immediately."
Van Gaal then went on to write about Guus Hiddink, who was appointed interim Chelsea boss after Mourinho's sacking.
"As you will know, my fellow countryman Guus Hiddink is in charge at Stamford Bridge until the end of the season. It came as a big surprise to see Jose Mourinho leave Chelsea, but we know how cruel football can be."
Van Gaal recalled captain Wayne Rooney to his starting XI for the match against Chelsea.
The England skipper started up front, replacing Marouane Fellaini, who dropped to the bench.
Morgan Schneiderlin, Bastian Schweinsteiger and Matteo Darmian also came in for Michael Carrick, Memphis Depay and Phil Jones.
Eden Hazard started up front for Chelsea in the place of suspended striker Diego Costa.
Louis van Gaal's Rooney call
Under-fire Manchester United manager Louis van Gaal has decided to restore captain Wayne Rooney to his side for Monday's match against Chelsea.
Rooney came off the bench for the second half of United's clash at Stoke on Boxing Day, but was unable to turn the tide and Van Gaal's men lost their fourth game in a row.
Midfielder Bastian Schweinsteiger returns from suspension for the game against the Blues and defender Matteo Darmian makes his return from a hamstring injury, but Jesse Lingard (hamstring), Marcos Rojo (shoulder), Antonio Valencia (foot) and Luke Shaw (leg) are out.
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