Harry Kane and Son Heung-min resigned the Gunners to another defeat as their struggles continue.
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.”
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