The side needed to win 4-0 to take the match to extra-time but were unable to score the crucial fourth goal despite leading by three goals at half-time.
The Frenchman was suspended for two matches by the football governing body earlier in the season, after the Gunner's Champions League qualifier against Udinese, but on that occasion to fine was less than £10,000 which was to be paid by Arsenal.
A statement on Uefa’s website read: “Arsenal FC manager Arsene Wenger has been suspended from carrying out his function for three UEFA competition matches by the UEFA Control and Disciplinary Body.
“Wenger has also been fined €40,000 (£37,000). He was facing a charge of improper conduct in relation to the UEFA Champions League round of 16 second leg against AC Milan in London on 6 March.
“The suspension applies to the next three UEFA club competition matches for which Wenger will be eligible.
“An appeal may be lodged against the verdict within three days of the dispatch of the reasoned decision.”
If the Gunners finish third in the Premier League this season then he will miss half of their group stage matches in next season's competition.