The loss meant Arsenal was eliminated at the third-round stage of the competition for only the second time in 26 seasons.
Arteta was back on the bench after a bout of coronavirus forced him to miss arguably their best performance of the season in the 2-1 defeat to Manchester City last weekend.
But he was not happy with what he saw at the City Ground, with substitute Lewis Grabban getting the winner for Forest, who were also the last team to beat Arsenal at this stage of the FA Cup.
Speaking after the game, Arteta told ITV Sport: "We are out of the competition and we have to apologise.
"When you have nine players out, that's a big explanation, but still I don't want to use excuses.
"The team we put out, I expected them to play better and to compete better than we did.
"When you don't do that in a cup against any opponent, you're out.
"What we've done today is nowhere near the standard required."
The 34th-minute withdrawal of left-back Nuno Tavares after a very sloppy start was the first indicator of potential frustration on Arteta's part.
Yet he opted against criticising the youngster, putting it down to a desire to simply change the level of the team.
Asked if it summed up his frustration, Arteta said: "No, it was a decision you have to make in certain moments to try improve the performance of the team, and that's what I did."
Arsenal will look to get this defeat out of their system as soon as possible, with an EFL Cup semi-final first leg against Liverpool and a north London derby trip to Tottenham on the horizon.