Ondrej Celustka struck 14 minutes from time to leave Inter reeling and pile the pressure on new coach Gian Piero Gasperini after only three games in the job.
The Turkish runners-up, only in the competition after Fenerbahce were withdrawn because of match-fixing allegations, looked comfortable throughout before Celustka expertly hooked the ball in on the slide after Halil Altintop hit the bar.
"We are not in a lucky moment, this defeats leaves us with lots of regrets but the team know they have given a lot and put in a decent showing," Gasperini told Sky TV.
"We are a team that will definitely get better, we have potential, will will get injured players back. The will and the determination of the players is clear. Moratti has always really supported me."
The coach has though already been criticised in the media by the club's owner Massimo Moratti, who has seen bosses Jose Mourinho, Rafa Benitez and Leonardo leave since their 2010 treble as last season went sour.
Former Genoa coach Gasperini, inexperienced at the top level, abandoned the three-man defence that flopped in Sunday's opening 4-3 Serie A defeat at Palermo and reverted to their traditional four at the back but it made little difference.
Wednesday's performance will reinforce that view and send odds tumbling of Inter going through a fifth manager in just over a year even if Trabzonspor failed to offer much threat until Czech Celustka's goal.
