The Gunners currently occupy the final Champions League place and are two points ahead of their north London rivals in fifth.
Despite Spurs having a game in hand on Arsene Wenger's men, the Poland international feels that they lack the class to finish above his side.
"Tottenham do not have enough quality [to finish fourth], so there you go," The Mirror quotes Szczesny as saying.
"If you look at Chelsea's fixtures, they have Tottenham and Everton left and they won't be easy games.
"If we win our next two games, it will be enough because either Chelsea or Spurs will drop points because they play each other."
Tottenham play Chelsea on Wednesday.