The Icelandic goalkeeper replaced No 1 Bernd Leno on Tuesday night with the move immediately proving costly for the Gunners and heightening the pressure on the Spanish manager.
After misjudging a cross to gift Gabriel Jesus the opener, Runarsson, 25, failed to keep out Riyad Mahrez’s free-kick, which slipped through his hands.
And Arteta has been forced to defend the decision, insisting he will “support” the £2m signing from Dijon.
“He hasn’t played a lot of games for us, he’s just adapting to the league,” said Arteta after the quarter-final exit. “We all make mistakes and we have to support him.”
While Arteta also maintained Leno needed a rest ahead of the festive period, with the Gunners facing London rivals Chelsea next up on Boxing Day.
“Yes, but as well we know that Bernd has played a lot of minutes,” Arteta added. “He needed some rest as well and we wanted to give opportunities to some other players.
“Alex has done well in the other games and these things happen in football.”
Arteta also described the challenge of defeating Pep Guardiola’s side as “impossible” after gifting City several goals.
Arteta said: “A really hard one to take, considering how the game went, the reaction the team had against this level of opponent.
“Obviously when you give the goals away that we gave, you make the game impossible.”
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