The Gunners were nine minutes from crashing out of the competition after falling behind to Wigan Athletic in their semi-final clash, but Per Mertesacker struck in the 81st minute to keep Arsene Wenger's men in the tie.
Arteta was one of the four Arsenal players to convert from the spot in their penalty-shootout victory over the Latics to give them a chance to win their first piece of silverware since 2005.
"We're really happy," Arteta told the club's official website. "It seems like at the moment we need to suffer and really earn things but I think we've done it today.
"We were unlucky to go a goal down the way the game was going but I think we showed a lot of character as well. We gave everything we had, we went forward, we created chances. It took us until the end to score and as for the penalties I think we were cool."
Arsenal will face Hull City in the final in May.