Roberto Mancini, who replaced Fatih Terim as Gala coach on Monday, enjoyed his return to Italy as his new charges dominated the Serie A champions in Turin, opening the scoring through Didier Drogba in the first half.
Juve did find a way back into the tie when Arturo Vidal levelled from the penalty spot 14 minutes from time and substitute Fabio Quagliarella thought he had grabbed all three points when making it 2-1 with time running out.
Bulut powered home a late leveller, though, guaranteeing Galatasaray their first point in the group while ambitious Juventus have only two after leaving Copenhagen, who lost 4-0 at Real Madrid in round two, on the back of a 1-1 draw on the opening night.
Drogba reminded Juventus why they had tried to sign him in January when he powered into the box to force a low save from Gianluigi Buffon.
Galatasaray tried again, unsuccessfully, through Bruma and in the 25th minute Juve suffered an early setback when Vucinic crumpled on the Turin pitch.
The Montenegrin was replaced by Quagliarella before Carlos Tevez slammed the ball against Gala defender Aurelien Chedjou.
Gokhan Zan then stepped across to block Kwadwo Asamoah's long-range effort and the Turkish side soon raced up the other end to opening the scoring.
Buffon was guilty of letting his guard down as he failed to deal with a quick Leonardo Bonucci backpass, Drogba stealing behind him to tap into an empty net.
Tevez missed by inches with a scorching effort as Juve looked to force an equaliser before the break.
But despite Asamoah going close and Quagliarella testing Fernando Muslera with a bullet header, Gala would maintain their deserved lead at half-time.
Stephan Lichtsteiner would not return to the pitch following the interval, Mauricio Isla replacing Juve's injured Swiss full-back.
The Bianconeri pushed forward in search of parity but Vidal could not find it from distance and Tevez's smart strike ended up in Muslera's gloves.
Quagliarella and Andrea Pirlo were wasteful with curling shots as the Italians continued to labour without reward toward the hour mark, and substitute Fernando Llorente was particularly guilty when glancing Asamoah's pinpoint cross over the bar.
Antonio Conte's men would get the chance to release some frustation in the 77th minute when Quagliarella tumbled in the box under the attention of Nordin Amrabat.
Vidal did not need much encouragement to step up to the spot and rifle Juve's equaliser under the bar with venom.
Conte began to urge his men forward and whooped on the touchline as Quagliarella appeared to turn the game on its head four minutes from time.
The skilful striker rose highest to nod home Pirlo's cross from the right and Juve's players streamed towards the stands to celebrate a victory salvaged from impending defeat.
It was not to be, however, as Gala wrested back a share of the points before the full-time whistle sounded.
Drogba was involved once again as Bulut swept up his headed pass to stun Buffon, and the home crowd, with an accurate finish that had Mancini smiling with relief.