The shock result meant that the Gunners slipped down to 10th in the table and fans at the game made their feelings known, booing the team off the pitch at full-time.
It is Arsenal's worst start to a season since Arsene Wenger took over as manager, but Szczesny believes that they are still capable of achieving a top four finish.
"Even after a game like that when everyone is disappointed, the fans are disappointed, the players are disappointed, it is very easy to give up and just say that we are not good enough," he told the London Evening Standard.
"We definitely believe we are good enough to do the job, we are going through a little bit of a rough time but just keep the faith and I'm sure sooner rather than later, we will get there.
"It is very hard right now. This defeat is even more disappointing because we played at home against a Swansea side in a game you should be winning comfortably when you play for this club."
The Poland international admitted that they were beaten by the better team on Saturday, saying: "We got battered and there's no hiding from it - they were the better team from start to finish and it makes it even harder for us.
"Of course, the confidence is knocked a little bit after a game like that but we are going to work even harder to make sure we get everything right on Saturday against West Brom because we just must win that game.
"It's about time for us to realise something has gone wrong and we have to do something about it because I don't think the fans deserved (what they saw).
"The stadium was empty when the game finished and I don't blame them. We must stick together and work harder because it was not good enough."
He continued: "We had a little conversation in the dressing room that was probably louder than usual. What happens in the dressing room stays in the dressing room but everyone was disappointed.
"I am not going to say we didn't want it enough but Swansea wanted it more and it showed. That's not good enough when you play for this club so I think before we start talking about qualities and what goes wrong tactically and stuff like that, you need to make sure everyone gives 100 per cent.
"There is time. If you look at the fixtures this month, we should be getting points. But we should have started on Saturday and we didn't which is very disappointing.
"Having said that, we've got a few games in front of us that we should be winning and we have to make sure we do that and climb up the table. To see Arsenal in 10th is just not right."