Arsene Wenger's side travel to face relegation-threatened Wolves on Wednesday where they will attempt to build on last weekend's 1-0 win over City.
That victory, secured by Mikel Arteta's late goal, effectively ended City's hopes of winning the English title and moved the Gunners to within 10 points of Roberto Mancini's side with six games remaining.
The primary target for the north London club is to finish third and gain direct qualification into the Champions League group stage, but Szczesny believes City may now falter, paving the way for Arsenal to clinch second spot with Manchester United all but assured of retaining their Premier League title.
"I believe we will finish third," said the Arsenal goalkeeper. "Man City have lost their chance of the title -- it was a big blow to us last year and they might drop a little bit, so who knows maybe we can catch them as well.
"Our aim is to win every game and if we do that we will be in the Champions League next season."
Wenger was frustrated by a mid-season slump that prevented his side from mounting a stronger challenge at the top of the table.
But a run of eight wins in the last nine league games has revived their season, with the unexpected defeat at Queens Park Rangers the only blemish on Arsenal's recent record.
And midfielder Yossi Benayoun believes they must maintain that run to fend off the challenge of fourth-placed Newcastle and Tottenham, two points behind the Gunners and Chelsea, who are four points adrift of Arsenal in sixth place.
"We had to take three points (against City) to open the gap," said Benayoun. "It will be very close until the end of the season, we have to continue to play like this," the Israeli added.
"At this time of the season I think every game is a difficult one. Teams will fight for their lives in this league and it is harder to play against them, especially when you play at their ground.
"It will be like this against Wolves. But if we are 100 percent committed, then we will win the game."
Wolves manager Terry Connor insists he has "no regrets" about taking on the position despite garnering a solitary point from his first seven games in charge after his promotion from assistant boss following the sacking of former manager Mick McCarthy.
Saturday's defeat by Stoke made it six successive losses for Wolves after they had drawn in Connor's opening match in charge at Newcastle.
They are now six points adrift at the foot of the table and relegation to the Championship is starting to look like a formality.
But Connor said: "I'm in charge and have been given a job to do till May 13 and will carry that through.
"That hasn't changed one iota for me unless Jez Moxey (the chief executive) or Steve Morgan (the owner) tell me otherwise.
"Wherever we are after that I am sure the club will sit down and take stock and decide where they go from there.
"I have no regrets about taking the job. It is a tremendous challenge and honour and privilege to be manager of Wolves in the Premier League.
"I am enjoying it -- but I just wish we could get a few results for the fans and the club."
Connor must decide whether to recall Roger Johnson and Kevin Doyle after leaving the pair on the bench for Saturday's defeat at the Britannia Stadium.