The 23-year-old found himself amongst the substitutes for a short period as Lukasz Fabianski was given a chance in the team.
Arsenal were linked with a move for a new keeper throughout the summer, eventually signing Emiliano Viviano on loan, and although Szczesny has been ever-present for the Gunners so far this season, he believes that any added competition is to be relished.
"The most important thing is the manager's faith in me. I'm not going to him and asking whether a new goalkeeper will be bought," Szczesny told Gazeta Wyborcza.
"I train very hard, pay him back with my performances. I'm not scared of competition."
Asked if he was surprised to find himself out of the team, Szczesny replied: "No. But only because he told me eight days before that second game with Bayern. I took this decision like a man.
"I had a hard time at Arsenal in March. I lost my place and had to fight my way back. I was given my chance, luck was on my side and Lukasz has got injured as well."
Real Madrid veteran Iker Casillas has been amongst the names linked with a move to the Emirates Stadium, something which Szczesny says he would welcome.
"Yeah, why not, we need as many great players as possible," he said.
The Poland international also spoke about the form of Manchester City goalkeeper Joe Hart whose displays for both club and country have come under much scrutiny in recent months.
"It's easy to judge by top tournaments. 18 months ago I have said that Joe Hart, Buffon and Casillas were the best," he said.
"Lately Joe was criticised, almost like English fans thought he would save every shot coming his way."