William Saliba cancelled out Raul Jimenez's first-half opener for Fulham before Bukayo Saka looked to have handed Arsenal a fifth straight win when he converted Gabriel Martinelli's cross in the 88th minute.
However, the England international's last-gasp strike was ruled out following a VAR intervention with Martinelli just offside in the build-up as the match finished 1-1.
Although Arteta's side reduced the gap to Liverpool from seven to six points, Arne Slot's men now have a game in hand after their match against Everton was postponed. Arsenal also failed to take advantage of Manchester City's 2-2 draw at Crystal Palace on Saturday.
"I am gutted that we didn't win it because we deserved to win from the beginning to the end," said Arteta.
"We did almost everything that we had to do to win it, and, but for millimetres we could have been sitting here with three points after a really strong and dominant performance against a really good team.
"You cannot cry about it, and that's the beauty of this league, too.
"We cannot feel sorry for ourselves. We have to stand up for ourselves, and now we have to continue to improve and look at what we can do better so that the opposition have zero chances to win the game. That's the objective."
Arsenal had bounced back from the international break with three straight league victories, and headed into Sunday's fixture fresh from recording a comfortable 2-0 win against Manchester United at the Emirates.
But Arteta's players looked lethargic in the opening period at Craven Cottage, with Saka registering Arsenal's sole shot on target in the first 45 minutes.
The Gunners improved after the break, and Saliba's strike from a corner - Arsenal's third goal in a row from a corner and their 23rd since the start of last season - looked as though it would pave the way for them to take all three points.
However, Arsenal were guilty of failing to create enough clear-cut chances, prompting Arteta to defend their record of completing two consecutive games without scoring a goal from open play.
"It's a great thing that we can again score from a set-piece," he added. "It's never enough if you don't score three, four or five.
"If we want to improve we have to be better in every action that we do. If they play 11 direct balls, I want to win 11. If they play the high-press six times, I want to play six times. We want to manage the game 100 per cent."
Following Jimenez's fine finish with just 11 minutes on the clock, the hosts set out to frustrate their opponents.
Arsenal - forced into another defensive rejig with Oleksandr Zinchenko joining Gabriel, Riccardo Calafiori and Ben White on the injury table - have taken just two points off Fulham in their last three clashes.
"It was a tough game as you would expect," said Fulham boss Marco Silva. "We knew Arsenal would be a difficult opponent but we showed great fighting spirit.
"The early goal was good from ourselves but Arsenal then took control of the game and I really believe that on the ball it wasn't our best game.
"When you are winning 1-0, the mindset of the players is not to take risks. They pushed us, and we didn't create many chances, but we created a difficult situation for Arsenal."