Salah opened the scoring in the fourth minute, running through from midfield and finishing calmly, before he turned provider to set up Noah Saviolo, who struck from inside the penalty area in the 39th.
Salah added a third in the 44th minute with a clinical finish from a Franculino Dju assist, sending the team he joined last month into the break three goals up.
He completed his hat-trick in the 80th minute, running more than half the length of the pitch unchallenged to score with a trademark side-foot finish.
Galatasaray coach Okan Buruk was sent to the stands in the 71st minute while Lesley Ugochukwu was shown a red card in the 87th.
German coach Thomas Reis took charge of Trabzonspor this week following the departure of Fatih Tekke earlier this month.
Trabzonspor are fifth in the table with 10 points from six games, three points behind leaders Galatasaray.