Paul Pogba added a crisp second in second-half injury time to increase the hosts' lead and put them in control of the tie ahead of the second leg on February 27.
Antonio Conte's side shrugged aside the memory of their last game against Turkish opposition, a 1-0 loss to Galatasaray that dumped them out of the Champions League in December, to take a two-goal lead into next week's second leg.
But Trabzonspor, who came into the game on the back of an abject run which has seen them fail to record a victory in 2014, will feel aggrieved after Olcan Adin's second-half strike was chalked off.
Carlos Tevez wasted the first chance of the game for the hosts in the fifth minute when he skied an effort high over Onur Kivrak's crossbar from close range.
Osvaldo made the Italian side's early pressure count in the 15th minute when he slotted the ball past Kivrak after a pass from Tevez rebounded off the face of a Trabzonspor defender and into his path.
On the half-hour mark Kivrak produced a decent stop to keep out an effort from Mauricio Isla, before Tevez curled a shot just wide from the left-hand side of the box.
Trabzonspor made a change early in the second half, replacing Emre Gural with Paulo Henrique.
And they were unfortunate not to be level in the 69th minute when Yusuf Erdogan set Adin up for a simple close-range finish, only for play to be pulled back after the official behind the goal adjudged the ball to have crossed the line for a corner before Erdogan played the pass.
With seconds left of normal time Pogba clipped the post with a deflected effort, before increasing the hosts' lead in added time with a composed curling finish after being set up by a neat cutback from Tevez.