Wenger's long-standing position at the north London club was in jeopardy after nine years without a trophy, before the FA Cup breakthrough in May rewarded the fans' patience.
Arsenal trailed 2-0 early against underdogs Hull City, before launching a brilliant come-from-behind win - eventually triumphing 3-2 in extra time.
Captain Arteta said Wenger's job was saved by their victory.
"He wasn't clear with his future," Arteta said, as quoted by the Daily Mail.
"He always said he wanted to stay but he didn't sign. In the semi-final - or even in the final - we never knew what was going to happen.
"If we lost the Hull game we'd be talking about a different story. That would have started a period of reflection and decision.
"It was a big game in Arsenal's history. We won the first trophy in nine years.
"You can go from being criticised to being hailed very quickly."
Alex Oxlade-Chamberlain has appeared in all five of Arsenal's Premier League outings in 2013-14, and Arteta said the England international has the most upside at the Emirates.
"The potential he has is second to none. If he really wants to, really focuses, he can go to the very top," Arteta said.
"He has the speed and skill to become an unbelievable winger, but he has the vision of a midfielder, and that's very difficult to find.
"Defensively, he needs to improve, but he has time and people to help him."
Arteta also leapt to the defence of much-maligned midfielder Mesut Ozil, who joined the club prior to the 2013-14 campaign.
The German midfielder cost Arsenal a reported £42.4million, but the former Everton man said the World Cup winner was "an artist".
"When you have players like Ozil with everything he involves - his price, his personality - people will either love him or not like him at all," Arteta said.
"When you're an artist, you're never going to perform nine out of 10 every week. It's much more difficult for a creative player to perform at that level.
"The opposition always try and stop you, so sometimes you don't shine. And then people ask: "Why don't you shine? You didn't do anything in the game.
"When that happens you get criticised, especially when it happens in big games. But you cannot deny Ozil is a special player: his movement, his calmness, his final ball is superb."
