Juve head coach Conte witnessed Pablo Daniel Osvaldo open his account for the Serie A leaders with a composed finish to break the deadlock at the Juventus Stadium in the first leg of the last 32 tie on Thursday.
Carlos Tevez then spurned several opportunities to double the lead against the Turkish outfit before half-time, including a chance shortly before the break where he showed neat footwork on the edge of the box but fired tamely at Onur Kivrak.
Juve were almost made to pay for their wastefulness with 20 minutes remaining when Gianluigi Buffon saved Paulo Henrique's effort, before Olcan Adin turned home Yusuf Erdogan's cut-back, only for the celebrations to be curtailed as it was adjudged that the ball had gone for a corner.
The hosts made the most of their reprieve as Paul Pogba curled home a second in injury time, but Conte was quick to concede that he would have preferred to travel to Turkey for the return leg next week with a bigger lead.
"We wasted a lot of chances in the first half, due to good goalkeeping and our own shortcomings, but did well to keep going and get the second," he is quoted as saying on Juventus' official Twitter account.
"Carlos Tevez had several chances, but above all made a lively contribution to the team performance.
"We wanted another goal and introduced a third striker. We did well in the end, hitting the post before grabbing our second."
Conte added on Sport Mediaset: "Trabzonspor had finished ahead of Lazio in the group phase, so were not to be underestimated.
"We did well, but there is still another game to play. We got too stretched out in the second half when pushing hard for another goal."
