The Gunners finally produced the required performance against a top-four Barclays Premier League rival to recover from the setback of going behind to an early goal from David Silva with a spirited second-half performance.
Midfielder Mathieu Flamini - who had netted a bizarre last-minute own goal in the 2-2 draw with Swansea on Tuesday night - equalised and German forward Lukas Podolski then came close to turning the match on its head when his angled drive was flicked behind by England keeper Joe Hart.
Szczesny believes it is the kind of display which can lead Arsenal on to a positive ending to the season, as Arsene Wenger's side aim to secure Champions League qualification once again and also a shot at FA Cup glory.
"We have not had a great time in the last couple of games, but it is all about responding well and that is what we did," said the Poland international, who shipped six goals in the heavy defeat at Chelsea last weekend which ruined Wenger's 1,000th match in charge.
"Of course it was difficult times, but we knew if we stuck together, carried on working properly then a result would come. Hopefully now we can kick on."
Szczesny added: "We owed it to the manager, the fans and everyone at the club, ourselves as well.
"We lost 6-0 (at Chelsea) and that is embarrassing for players who represent this club.
"You want to prove to yourself and prove to everyone around you that you are better than that.
"We have still got a lot to play for, the FA Cup and we are in a decent position in the league, so we will be on our toes and at our very best to go on and win all the games.
"I am confident we can have a very good run-in and finish strong."
Gunners boss Wenger hopes his squad are now finally through their "nightmare" of two results which could have derailed what has largely been a promising campaign in the wake of more questions over the character of the side.
"It has been very bad," said Wenger, "but as well very positive because it is always an opportunity to show you have some mental strength, and so I am very proud of my players because they have shown that.
"We have gone through an absolute nightmare, a storm this week.
"It is difficult to protect the players from that, the only way you can deal with it is to respond on the pitch.
"The critics come very quick and very hard, the only way we can deal with it is to respond in the next game and that is what we did."
Wenger - who on Friday hit back at criticism of his side's qualities by former Manchester United midfielder Paul Scholes - added: "They (players) have to let people talk. I enjoy football and don't want to spend my time responding to everybody because I am long enough in the job to know that is not the most important thing."