The 34-year-old Egypt forward left Liverpool on a free transfer this summer after scoring 257 goals for the club across nine trophy-laden years, and had initially been expected to move to Saudi Arabia before instead agreeing a two-year deal with Trabzonspor worth a reported £280,000 a week.
Sunday's match was his third appearance for the club, following a substitute cameo against Kasımpaşa and a start in Trabzonspor's Europa League play-off first-leg defeat to Ferençvaros.
Salah opened the scoring in the eighth minute, reacting quickest after Başakşehir goalkeeper Muhammed Şengezer failed to hold a routine catch, with his effort deflecting in off the post.
Eldor Shomurodov levelled from the penalty spot in the 19th minute after a VAR review for handball, but Salah restored Trabzonspor's lead three minutes into the second half, cutting in from the right and exchanging passes with Ernest Muçi before finding the far corner with a first-time finish.
The Egyptian was shown a yellow card for simulation with 11 minutes remaining, and thought he had completed his hat-trick in stoppage time when he found an empty net from 45 yards out, only to be ruled offside. He led all players on the pitch for shots, shots on target and chances created during the match.
Salah's opening goal made him the third Egyptian player to score for Trabzonspor in the Süper Lig, following Ayman Abdel-Aziz and Trezeguet, while his second made him the first foreign player to score twice against Başakşehir for the club.
The result secured Trabzonspor's first league win of the season, moving them to four points from their opening two matches.