United went into the break behind after Sinan Gumus and Bruma had cancelled out Ibrahimovic's opener, but second-half strikes from Wayne Rooney (2), Marouane Fellaini and Juan Mata saw the Red Devils come from behind to record a comfortable win.
Ibrahimovic was named in the starting lineup just days after beginning training with his new club, and it took the 34-year-old less than four minutes to open his account in a United shirt.
The Swede did it in trademark fashion in front of his home fans too, producing an acrobatic scissor kick from Antonio Valencia's cross to hand his side an early lead.
It was a dream start for both the player and the club, but United failed to build on that and Galatasaray soon began to settle into the match as the first half wore on.
Goalscoring chances continued to be few and far between, but De Gea was the busier of the two keepers, being forced to deny one-time United target Sneijder's long-range effort before keeping out a Yasin Oztekin header from the resulting corner.
Rooney had a rare effort for United towards the end of the half, but the Premier League side found themselves behind just a minute later when Bruma played a one-two with Gumus before tucking his finish into the bottom corner with aplomb.
Ibrahimovic's debut lasted just 45 minutes as he was one of five players to make way at the interval, but that didn't stop United from levelling things up 10 minutes after the restart when Rooney, appearing in his first match of pre-season, steered Valencia's cross past the keeper.
That sparked a spell of three goals in seven minutes, and Rooney doubled his personal tally just three minutes later when he drilled a penalty down the middle after Marcus Rashford had outpaced the defence and been felled in the area by Fernando Muslera.
That restored United's one-goal advantage, but they soon extended the cushion when Valencia stood the ball up to the back post for Fellaini to loop his header over the keeper and into the far corner.
Galatasaray looked like a beaten team in the closing stages as they waited for the final whistle, and they avoided any further damage as United cruised to the first win of their pre-season tour.
The Red Devils will next face Everton in Rooney's testimonial before taking on Leicester City in the Community Shield next weekend.