It means they remain in the bottom half of the season while their rivals return to the summit after their derby success.
Arteta saw his side perform impressively in the second-half especially, but a lack of cutting edge meant they were unable to overcome the two-goal lead Tottenham had establish at the break.
Pressure on the Spaniard will only mount but Mourinho conceded his side were tested in the second-half and has backed Arteta to bounce back.
He said: “It was a big game in every sense, not just because it was Tottenham-Arsenal.
“I want to give words, congratulations to Mikel Arteta because he gave us a very difficult game. Tactically they are very good, very organised.
“They gave us problems, problems we were able to resolve. They have a good tactical courage and incredible spirt. They are a good team and he is a good coach.
“We were reading the game and making the changes to win it because 2-0 was in our hands against a great and difficult opponents but my boys did well today
“I am very happy with the result and the performance but of course I want the team in the second half to play in a different way.
“In reality you don’t always do what you want to do because your opponent forces you in a different direction.
“You are winning 2-0, why would you expose your defensive block if you don’t have to? It was a very difficult game.
“I can imagine tomorrow the headlines will about them not being in a good position in the table but I believe with these players and Mikel it will be Arsenal v Tottenham again.”