Pablo Daniel Osvaldo's first goal for the club put hosts Juve in front on 16 minutes before a controversial moment 20 minutes from time.
Trabzonspor substitute Henrique had a shot saved by Gianluigi Buffon and Olcan Adin scooped home Yusuf Erdogan's cut-back, but was denied as the officials decided the ball had gone out for a corner.
Paul Pogba's goal in stoppage time then added insult to injury for the Turkish side.
Fiorentina will also be heavy favourites to progress along with Juve into the last 16 after a 3-1 win over Esbjerg in Denmark, while Napoli drew 0-0 with Swansea City at the Liberty Stadium.
Alessandro Matri opened the scoring for Fiorentina, only for Martin Pusic to equalise immediately, before the Italians took control through Josip Ilicic and an Alberto Aquilani penalty.
Dutch giants Ajax were swept aside 3-0 at home by Salzburg, who have now scored in 63 consecutive matches, a run that stretches back to December 2012. Sadio Mane and Jonathan Soriano's brace did the damage in Amsterdam.
A one-minute silence was observed at all games involving Ukrainian sides out of respect to those who have lost their lives during the recent political unrest in the country, with some Shakhtar Donetsk players in tears ahead of their 1-1 draw with Viktoria Plzen.
Back in Ukraine, Yevhen Konoplyanka's late penalty gave Dnipro a 1-0 win over Tottenham, strikes from Eduardo Vargas and Antonio Barragan saw Valencia defeat Dynamo Kiev 2-0 and the game between Chornomorets and Lyon finished goalless.
Last season's runners-up Benfica have the advantage heading into their second leg with PAOK, returning from Greece with a 1-0 victory thanks to a second-half goal from Rodrigo Lima.
Bulgarian outfit Ludogorets shocked Lazio 1-0, Roman Bezjak's strike securing a famous victory in Rome before Luis Pedro Cavanda and Svetoslav Dyakov were dismissed in the second half as both sides finished with 10 men.
Sevilla drew 2-2 in Slovenia against Maribor, while Porto's match with Eintracht Frankfurt finished in the same result after the German outfit came from two goals down to give themselves a healthy chance of progressing.
Nick Viergever's strike a minute from time earned AZ a 1-0 win at Slovan Liberec, while Anzhi and Genk played out a 0-0 draw in Russia.
Elsewhere, Real Betis drew 1-1 at home with Rubin Kazan and Maccabi Tel Aviv's game with Basel ended goalless.
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