The on-loan Southampton attacker netted just 16 minutes into the game in Turin, while the France international fired home in the dying seconds to leave the visitors devastated
The on-loan Southampton struck just 16 minutes into the first-leg clash in Turin, coolly slotting home after the ball had fallen nicely for him.
Olcan Adin controversially had a goal disallowed midway through the second half and Juventus took full advantage, with Paul Pogba firing home in injury time to leave the visitors crestfallen.
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 Argentine scooped well over the crossbar.
Despite that miss, Tevez played a part in the opening goal 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 Onur Kivrak.
Juve 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 Bianconeri No.10 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 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 Gianluigi 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.