Pablo Daniel Osvaldo marked his first Juve start with the opening goal in the first leg encounter at Juventus Stadium, in a match Antonio Conte's side dominated.
But there was controversy in the 70th minute when substitute Henrique had a shot saved by Gianluigi Buffon, and Oldca Adin scooped home Yusuf Erdogan's cut-back but was denied as the officials decided the ball had gone for a corner.
Osvaldo broke the deadlock on 16 minutes with a calm finish past Onur Kivrak and Juventus continued to dictate the tempo after the break, but were thankful Adin's goal was ruled out as they struggled to put the game to bed.
However, Paul Pogba scored in stoppage time as Juventus inflicted a first Europa League loss of the season on Trabzonspor, and take a comfortable lead to Turkey for the return leg next week.
The home side made a bright opening and orchestrated the first opportunity after five minutes when Mauricio Isla picked out Carlos Tevez on the edge of the box but the Argentina international scooped well over the crossbar.
Despite that miss, Tevez played a part in the opening goal on 16 minutes as his pass to the wing ricocheted kindly into the path of Osvaldo who kept his cool to open his Juventus account by slotting under Kivrak.
Juventus continued to attack and spurned two opportunities to extend the lead shortly before the half-hour mark. First, Tevez slipped in Isla who cut inside and saw his close-range effort saved by Kivrak, before the Argentinian curled wide inside the box when perhaps he should have hit the target.
Tevez was proving a constant menace to the visitors' defence, though he was unable to test Kivrak fully after neat footwork on the edge of the area.
The chances continued to pile up for Juventus after the break, with Paul Pogba dragging wide from 20 yards and Osvaldo curling an effort from the left that Kivrak easily dealt with.
Tevez had another tame effort saved on the hour mark, but he remains without a goal in continental competition since April 2009.
Trabzonspor thought they had levelled on 70 minutes. Substitute Henrique forced Buffon into a save, and Adin turned in Erdogan's assist, only to see his celebrations curtailed when it was adjudged the ball had crossed the line for a corner.
Pogba clipped the post in the closing stages, before curling home Tevez's centre in stoppage time to add gloss to the scoreline.
