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Arsenal beat Bournemouth 2-0 with goals from Gabriel and Mesut Ozil to move on to 39 points from 19 Premier League games, one ahead of Leicester City who are at home to Manchester City, in fourth, on Tuesday.
Dutchman Van Gaal, whose tenure in the Old Trafford hot seat is under intense scrutiny after a Champions League exit and three successive league defeats, was afforded a warm reception by fans who have been critical of his team's lack of flair.
Van Gaal hinted after a 2-0 defeat at Stoke City on Saturday that he could walk away but, after a performance of attacking intent against Chelsea, he said he would not quit.
"No reason to resign for me, maybe the media want that but I shall not do that," he told BT Sport. "You have seen they (the players) are fighting for me.
"Even after this result they (the fans) were applauding. I have full confidence in the board and more importantly, in my players."
United, now eight games without a win in all competitions, struck the woodwork twice in the opening 16 minutes. Juan Mata rattled the crossbar with a fierce left-footed effort and Anthony Martial drilled the ball against the post.
But it was not all one-way traffic. Keeper David De Gea kept out John Terry's powerful header and made a fine double save to deny Pedro and Cesar Azpilicueta at the start of the second half.
Opposite number Thibaut Courtois then somehow kept out Ander Herrera's close-range shot.
Chelsea, playing their second game under interim manager Guus Hiddink after Jose Mourinho's sacking, wasted the game's best opportunity when Nemanja Matic raced clean through before blazing the ball wildly over.
Wayne Rooney might have won it late on but skied the ball over to leave United in sixth spot on 30 points.
Champions Chelsea, who were minus the suspended Diego Costa and played without a recognised striker, are 14th with 20 points.
Van Gaal bemoaned United's luck in front of goal.
"It was a very good performance but you have to score ... we don't have the luck we need," he said. "We were the dominant team."
The draw left Hiddink reasonably content.
"On one hand I am happy with a point because we had a difficult game but, on the other, we had two big chances to get three points. In general I'm happy," said the Dutchman.
Arsenal, thrashed 4-0 at Southampton on Saturday, got back to winning ways by seeing off Bournemouth at The Emirates.
Ozil set up centre back Gabriel for the opening goal on 27 minutes, his first for the club, and scored the second after the break by sidefooting home Olivier Giroud's flicked pass.
Tottenham Hotspur rose to third after Son Heung-Min netted a late goal in a 2-1 victory at 10-man Watford.
Spurs went ahead through Erik Lamela at Vicarage Road. In-form Odion Ighalo levelled with his 14th league goal of the season before South Korean Son flicked the ball past former Tottenham keeper Heurelho Gomes in the final minute.
Everton and Stoke served up a seven-goal thriller at Goodison Park as the away side came from 3-2 down in the last 10 minutes to win 4-3 with goals from Joselu and Marko Arnautovic's stoppage-time penalty.
Romelu Lukaku's double for Everton moved him level with Leicester's Jamie Vardy at the top of the scoring charts on 15 goals while Stoke's record signing Xherdan Shaqiri struck his first two goals for the club.
Substitute Andy Carroll's 79th-minute winner lifted West Ham United to seventh with a 2-1 victory over Southampton while Darren Fletcher's goal for West Bromwich Albion was enough to beat struggling Newcastle United 1-0.
Aston Villa's miserable season continued with the bottom club, marooned on eight points, going down 2-0 at Norwich City.
Crystal Palace and Swansea City drew 0-0.