Juan Mata and Anthony Martial both hit the visiting team's woodwork inside 20 minutes as United made a much more adventurous start to the match than has been apparent so far this season.
However, United has not won since November 21 -- a run of eight games without victory -- and is nine points adrift of leader Arsenal at the halfway stage of the 2015-16 English Premier League season.
Arsenal claimed top spot with a 2-0 victory at home to Bournemouth, bouncing back from Saturday's 4-0 drubbing at Southampton.
While van Gaal lamented his side's missed chances, he took heart from the team's performance.
"Will I resign? On the contrary," the 64-year-old told UK match broadcaster BT Sport. "When the players can give such a performance with a lot of pressure, there is not any reason to resign.
"Maybe the media wants me to, but I shall not resign. It's not a question of staying, it's a question of fulfilling my contract.
"Even after this result the fans were applauding, so I cannot say I'm very much concerned. In this football world you never know, but I'm fully confident in the board and my players."
Van Gaal had said his players were afraid of expressing themselves due to the pressure mounting on the team, having been knocked out of the European Champions League at the group stage and most recently losing to Stoke on Boxing Day.
But Spain star Mata almost made the perfect start when he rattled the crossbar in the third minute.
United goalkeeper David de Gea then did well to block John Terry's close-range header before Martial hit the inside of the post at the other end, cutting inside Branislav Ivanovic to drill a low shot which left Thibaut Courtois beaten.
The Chelsea goalkeeper did well to stop Ander Herrera breaking the deadlock in the 56th minute, making a reflex save from the Spaniard, and soon after the London team's Brazilian midfielder Willian was lucky not to concede a penalty when he handled the ball under no pressure from the opposition inside his area.
United's courage seemed to falter the longer the game stayed goalless, and Chelsea should have scored just after the hour mark when Serbian midfielder Nemanja Matic wasted a golden opportunity as he broke clear on the counter-attack.
Captain Wayne Rooney, back in United's starting lineup, also squandered a late chance as he volleyed high and wide -- and the England forward was lucky to get just a booking for a rash studs-up challenge on Oscar in time added on.
Van Gaal's fellow Dutchman Guus Hiddink is now unbeaten two games into his second spell as Chelsea's interim manager, with the Blues moving up to 14th in the table.
Arsenal secured top spot for at least 24 hours as Mesut Ozil was involved in both goals against a Bournemouth team that dropped to 16th, on goal difference below Chelsea and Norwich.
Brazilian defender Gabriel Paulista headed home the German's first-half corner, and the midfielder sealed victory after the break when he combined with Olivier Giroud.
This season's surprise package Leicester can regain the lead with victory over fourth-placed Manchester City on Tuesday.
Tottenham moved up to third on goal difference with a 2-1 win at 10-man Watford, as South Korean forward Son Heung-Min netted an 89th-minute decider.
Fifth-placed Crystal Palace stayed a point ahead of Manchester United after drawing 0-0 with Swansea, while West Ham moved up to seventh as substitute Andy Carroll scored the winner in a 2-1 victory against Southampton.
In-form Stoke followed up the win over United with a 4-3 success at Everton, as Xherdan Shaqiri scored twice before Marko Arnautovic fired a late decisive penalty after being fouled.
Aston Villa stayed bottom after losing 2-0 at Norwich, while Newcastle also remained in the relegation zone following a 1-0 defeat at West Brom.