Salah, who joined Trabzonspor on a free transfer after leaving Liverpool following a trophy-laden nine-year spell, also had two goals disallowed in a standout performance.
Making his first league start after appearing as a substitute in Trabzonspor's season opener, Salah put his side ahead in the eighth minute, capitalising after Basaksehir goalkeeper Muhammed Sengezer failed to hold onto the ball.
Basaksehir's Eldor Shomurodov equalised from the penalty spot in the 19th minute, but Salah restored Trabzonspor's lead three minutes into the second half, bursting down the right, exchanging passes with Ernest Muci and slotting the ball into the far corner.
The 34-year-old winger had a stoppage-time effort chalked off for a foul in the build-up and another ruled out for offside when he found the empty net from long range with the goalkeeper up-field for a corner.
He will now turn his attention to Thursday's Europa League playoff second leg away to Ferencvaros after his side suffered a 1-0 home defeat in the first leg.