The Gunners could have won it, too.
A debatable handball in the 95th minute gave Arsenal a penalty, which Mikel Arteta, surprisingly stepped up to take ahead of Olivier Giroud. His low drive to the right corner was well saved by Mark Schwarzer, and a breathtaking game ended with a flourish.
There will be many questions asked of Arsene Wenger and Arsenal, as they are off to their worst Premier League start in three decades. They need to get themselves in gear soon, or it will take something historic to make the top four.
Let's take a look at six things we learned today.
Maybe other teams don't just magically play better when they face Arsenal. Maybe Arsenal just makes them look better with their own poor performance.
We can't be sure, but the same thing keeps happening every single week.
Fulham certainly looked on their game today, though. A combative, if occasionally lazy, midfield kept much more possession than they were expected to. That let them feed the consistently threatening and occasionally excellent striking duo Bryan Ruiz and Dimitar Berbatov.
At times their consistently good play made Arsenal look average. And since Arsenal are a mid-table club right now, perhaps they are just that.
Out of my goal, Vito!
I've never really been as high on Vito Mannone as some, nor as low.
After all, the man started the season as Arsenal's third-choice goalkeeper. He has done fairly well since Wojciech Szczesny's ankle injury against Southampton.
Now it's time for him to go.
There are certain saves that a keeper for a top team (or at least one that wants to be) has to make, and Mannone can't make them. For example: Alexander Kacaniklic's downward header from about 12 yards out never should have gone in.
Vito's done well, but his time is up.
Those who accuse Dimitar Berbatov of being lazy just can't appreciate the nuances of the most subtly classy footballer in the world. Berbatov might be an enigma and an unusual striker, but he's absolutely the best player on his team.
His first touch is velvet.
He glides past players in either midfield or up front with seemingly little exertion. He lulls you to sleep and then finishes you off with a clinical touch.
He is extremely clutch.
Berbatov's first goal was partly due to poor defending by Bacary Sagna and partly due to opportunism. Sliding behind Sagna, he avoided everyone's attention before neatly tucking his header in the back of the net.
Mikel Arteta could use a lesson or two about penalty-taking as Berbatov calmly slid the ball past Vito Mannone to give Fulham a lead that they would quickly surrender.
I've never been one of Olivier Giroud's doubters.
Even during his barren run to start his Arsenal career, I knew that his unusual combination of physique and technique would make him a worthy central striker.
Can you imagine anyone else in the team scoring the two goals he did today?
Certainly not Theo Walcott, who is chomping at the bit to take Giroud's place. In his best game of the season, the Frenchman continued his fine goalscoring form and almost willed Arsenal to a victory.
Both headers, especially the second, were pure class.
He also lashed a shot from distance that Mark Schwarzer barely saved and a decent strike early on from even further out that barely skated over the bar.
As Olivier Giroud gets better and better, there is less of a case for anyone else on the team to replace him.
We know by now that while statistics are often helpful in evaluating players and teams, in football, they cannot possibly tell the entire story.
If they did, how could the best defence in the Premier League look so terrible?
Arsenal were frequently cut apart by a team that is their equal according to the only stat that matters: points.
The first goal was the result of a switch-off by Bacary Sagna. Alexander Kacaniklic should not have had a free header to score the second. And Mikel Arteta conceded the penalty that led to the third because no other defender was in the box to cover.
The main point is this: If you concede three goals to a team whose first-half play was at times lazier than a drowsy Andrey Arshavin, you have some serious problems.
And you're just not that good.
Here we are. Arsenal have not started a season this poorly since 1982, despite having players who are undoubtedly quality.
Why are the Gunners in this position?
I'm not the manager. I don't know what Arsene Wenger tells his players after a game like this, or what is done in training to rectify problems that have existed for years.
We thought earlier in the year that the defence was improved—despite statistical evidence to the contrary, it hasn't. We thought that this team could be consistent against teams of all levels—it hasn't been. We thought that Arsenal could challenge for silverware—they clearly can't.
Remember how terrible we felt about Arsenal's start last season? One year later and the team is worse.
Fans will keep paying the highest ticket prices in the world to see a mid-table team, but unless Arsenal show us anything different, they cannot pretend to be anything else.