Arsene Wenger's side will face Swansea City at the Emirates Stadium on Wednesday evening in a replay of their third round tie, although they have already lost to the Swans at home in the Premier League this season.
Szczesny told the club's official website: "We have already lost against them at the Emirates but we managed to get a draw [at the Liberty Stadium].
"I hope we can qualify for the next round because we realise the FA Cup could be a massive chance to win a trophy this season.
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“To not win anything for seven years is not good enough; any trophy would do but we realise how important the FA Cup is for the fans. We would love to do it."
In Michu, Swansea have one of the most in-form forwards in the Premier League and Szczesny knows that the Arsenal back line will have to be at their best to deny a player who has already scored past them three times this season.
"He is as good as they get in the Premier League," the goalkeeper observed. "Paying £2 million for him must be the bargain of the season, probably one of the best ever in the Premier League.
"I think he has scored 15 or 16 goals and we are only in January, so I have been very impressed with him.
"I am not scared of him but he has been fantastic. He seems to be getting on the end of everything at the moment."