The Gunners recovered from the setback of going behind to an early goal from David Silva with a spirited second-half performance as Mathieu Flamini's strike earned them a crucial point.
Szczesny believes it is the kind of display that Arsene Wenger's side need as they bid to secure Champions League qualification once again and also lift the FA Cup.
"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 conceded six goals in the 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.
"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."