Chances were at a premium between the two sides in a match which failed to live up to its hype. The result leaves Chelsea top of the table by a point.
United could well have left north London with victory but for a brilliant save from Wojciech Szczesny, who pushed a header from Robin van Persie onto the crossbar.
United, knocked out of the FA Cup in round three, now have a training break in Dubai before they resume Premier League action at Crystal Palace on February 22.
"All I can do - and I keep saying it - is we will try and win the next game and pick it up," said Moyes.
"If there is one club in history who have been great at winning games in the second half of the season and putting pressure on the teams above it is Manchester United. We will continue to try and do that and make it work.
"We can only do our job the best we can. We want to play better. We cannot bother about other teams. We can only put pressure on by winning our games."
Moyes added: "I believe the desire is still there in the squad.
"I see them and they are great lads. They are fantastic with their training and their commitment is first class.
"You saw how good Vida (Nemanja Vidic) was and how good Rio (Ferdinand) was when he came on. They are a great group of players and a very experienced group of players. They have been winners all of them.
"They all have medals and when you have won you want to win more. They are all still hungry and determined to be successful."
Moyes felt on another night, United could have left north London with all three points and added: "I thought we did a lot of good things in the game. We had good shape and were nice and compact. We had some good moments in the game.
"We had a great chance in the opening minute. I thought whoever would have got the first goal it would have been important and we had a great opportunity to get that.
"It (Van Persie's chance) was a great move, a great ball from Wayne Rooney and a great header. The goalie made a great save and near enough pulls it out of the back of the net.
"On another night Robin might have had a hat-trick. He had three good chances."
Opposite number Arsene Wenger was pleased with the response from the 5-1 trashing at Liverpool last weekend, if somewhat frustrated by the final result.
He said: "It was important not to lose, but it was also important to win. I felt we were nervous.
"Our team was highly focused not to concede, and that maybe restricted our game going forward a little bit because we were certainly hit by the five goals we conceded on Saturday, and you could feel that here."
Arsenal now host Liverpool in the FA Cup fifth round on Sunday and then tackle Bayern Munich in the last 16 of the Champions League.
Wenger said: "(It was) first (about) not losing tonight, then (we can) come back to our normal level offensively.
"We have come out two difficult games and are still in a strong position, that is the only positive."