Captain Robin van Persie netted twice from the penalty spot as the Gunners beat Aston Villa 3-2 to book their place in the fifth round.
Richard Dunne and Darren Bent had put the visitors in command at the break.
However, Arsenal were given a lifeline at the start of the second half when Dunne upended Aaron Ramsey, and Van Persie coolly dispatched the resulting spot-kick.
Within two minutes, Theo Walcott had levelled via a fortunate ricochet and then another penalty from the Gunners skipper completed the turnaround to send Arsene Wenger's men through to a last-16 tie with Sunderland or Middlesbrough.
Wenger said: "I was very impressed because I felt that when you come out of a bad period like we are, and you are 2-0 down, you can feel sorry for yourself or you can show you have the mental qualities to play for this club.
"I was very proud the team came out playing on all cylinders and showed they are not ready to lie down."
The Gunners boss added: "That is was a good test to show that we had the mental qualities to come back. On the other hand it was important that we didn't panic.
"To concede a goal just on half-time for 2-0 was a difficult blow, but we responded very well against Manchester United here and we did it again today [Saturday].
"Overall it was a classic FA Cup tie in England: 2-0 down and coming back, we had to fight to score three goals and we did it well."
Wenger had been given a rough ride by some home supporters following last weekend's 2-1 defeat by United, which was a third straight in the Barclays Premier League to lose ground on the top four.
The Arsenal manager said: "I felt they were quite positive [today] compared to last time.
"I am sad we cannot always keep them happy, the only thing we can do is show the spirit that we showed today [Saturday].
"I just want to do well for this club that I love. I understand that people question my decisions, but I wouldn't understand if they questioned my commitment and my dedication.
"The rest I understand, that sometimes people are not happy with my decisions."
Wenger believes this result could transfer through to the league.
"I hope that it convinces the team we have the required qualities and we can transfer that to the Premier League," he said.
Wenger is also remaining positive over Jack Wilshere's fitness.
The 20-year-old was first troubled by an ankle problem while on international duty with England last summer and, apart from a pre-season appearance in the Emirates Cup, has not played a competitive match for the Gunners this season.
Wilshere, who needed surgery after initially wearing a protective boot, was set to return in early February, but felt pain when running in training earlier this week and will now have a series of consultations with specialists before deciding how to proceed.
The fitness of the combative midfielder will also be of concern to England boss Fabio Capello as he prepares for his side's Euro 2012 campaign in Poland and Ukraine this summer.
But Wenger revealed that it could be only a minor setback.
"We have no real news yet. The first news we have had is better than expected. He is not in a boot, but I don't know more," said the Arsenal boss.
"His morale looks quite good and I hope it will only be a short setback. There will be more tests this week."
Wenger added: "I do not know yet when he will start training. I would not like to make any mistake in my statement because I did not speak to the medical people today [Saturday]."
Villa boss Alex McLeish could not hide his frustration at failing to hold off Arsenal's fightback.
He said: "I am very disappointed we did not see it out and it is a missed opportunity.
"The fans will be gutted at seeing us let that lead slip away.
"I said to the lads at half-time that I would like to be congratulating them at full-time, because it was an international performance in the first half.
"The penalty gave them a huge lift, and we looked a bit leggy, but we went right to the end and could even have had an equaliser with a late couple of chances, but it was not to be.
"We have to learn from this and try to obtain that winning mentality."
McLeish felt centre-back Carlos Cuellar had been caught in the face by a stray elbow from Van Persie during the second half.
The Villa manager said: "I thought it was a clear elbow, but the referee did not seem to think there was anything in it.
"I thought he put the elbow towards Carlos' face. The big man is not a guy to lie down; he got up pretty quickly."
McLeish added that he would like the FA to look at it, saying: "Yes, if it was an elbow in the face - look at my face and my nose is a result of elbows all my career. It is something which is dangerous.
"I never mentioned it to the referee. I saw it in real time.
"I don't like to point the finger at players, but it has happened to mine enough times."