The pair were charged with using "inappropriate language" towards Swiss referee Massimo Busacca following the Gunners' Champions League last-16 second leg defeat to Barcelona earlier this month.
Both Wenger and Nasri were angered by Busacca's decision to show Robin van Persie a second yellow card at a time when Arsenal were leading the tie on aggregate, a decision they felt cost their side the match.
UEFA subsequently handed out one-match suspensions to the French manager and his midfielder, while Wenger was also hit with €10,000 fine.
The Arsenal boss had originally insisted he would staunchly deny the charge and called on UEFA to show some "humility'', while comparing the organisation to a "dictatorship''.
The Gunners could have lodged a formal appeal, but the matter will now not be taken any further.
"Arsenal have decided not to appeal against UEFA bans for Arsene Wenger and Samir Nasri," a statement on the club's official website read.
"The manager and the midfielder were each suspended for one European match following charges of improper conduct in the wake of Arsenal's Champions League exit at Barcelona. Wenger was also fined 10,000.''