Arsene Wenger's side beat the Capital One Cup holders with goals from Nacho Monreal and Gervinho to keep the pressure up on fourth-placed Chelsea in the hunt to feature in Europe's elite competition next season.
Oxlade-Chamberlain, who impressed throughout the game on Saturday, believes that his side has to build on the confidence gained from beating Bayern Munich and Michael Laudrup's side to finish the season as strongly as possible.
"Everyone in the Premier League feels pressure to get results," the 19-year-old winger told the club's official website. "It is part of being a footballer. We have to take that in our stride because we know we have the characters to deal with it.
"The Swansea result is a good start but we can't look back, we have to look forward at what we can achieve. That will be important for the run-in so it is a matter of taking it step by step.
"We knew it would be a hard game but we had to take everything from Wednesday’s win at Bayern and use it in this game. We have always had belief in ourselves."
Despite a number of disappointing setbacks this season that will leave Arsenal trophy-less for an eighth season in a row, Oxlade-Chamberlain says the players must focus on each game at a time rather than worry about their rivals.
"Obviously there have been some hard patches," Oxlade-Chamberlain said. "But wins like the one at Swansea affect the confidence in a positive way. We have to take that into the next game and look for three points again.
"It is going to be tough. We just have to stick to what we know. There are a lot of teams around us who we can't control. We can only control ourselves and well see what happens at the end."