The Black Sea Storm found the first goal late against the Black Eagles with midfielder Abdulkadir Omur in the 49th minute, just before the end of the first half, at the Vodafone Park in Istanbul.
Former champions Besiktas equalized the score as Canadian forward Cyle Larin scored in the 62nd minute.
But Trabzonspor crushed their opponents' hope for a draw with Danish attacker Andreas Cornelius' late goal in the 96th minute.
During extra time, Trabzonspor goalie Ugurcan Cakir was shown a second yellow card and sent off with a red in the 102nd minute, and the game ended 2-1.
Trabzonspor remain undefeated and have solidified their place at the league's top spot with 30 points, while Besiktas are ranked 4th with 20 points.