Arsenal ended their nine-year trophy drought with a nail-biting 3-2 victory over Hull City after extra-time in the FA Cup decider at Wembley in May.
Arsene Wenger has since recruited Alexis Sanchez, Mathieu Debuchy, Calum Chambers and David Ospina for the 2014-15 Premier League campaign.
Those arrivals in London have made Arsenal one of the favourites for the title, but Szczesny is not feeling any pressure.
"I don't think there's pressure at the moment. We took a lot of that pressure off by winning the FA Cup at the end of last season and had a very good game in the Community Shield against Man City," the Pole said.
"I think we're enjoying our football and can't wait to get started.
"There's not too much pressure on winning the title yet, we're not thinking about that. We just want to kick off the season well and win the first game."
Arsenal's Premier League season kicks off at Emirates Stadium on Saturday, when they host manager-less Crystal Palace following Tony Pulis' shock departure.
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