However, it could have been even better for the Turkish side, who missed a first-half penalty and still have it all to do to dislodge their opponents from second place in Group H and progress to the knockout stages.
It was a frustrating night for the hosts, with Dany Nounkeu putting through his own net to give Cluj a 19th-minute lead before Felipe Melo missed the penalty 10 minutes before half-time.
Galatasaray dominated the second half and got their reward through Burak Yilmaz 13 minutes from time but they could not force a winner.
Galatasaray started the game without a point from their opening two matches against Manchester United and Braga, making tonight's encounter in reality a must-win for the Turks.
Nicolas Godemeche shot over after just 19 seconds and Rafael Bastos' effort went wide less than a minute later as Cluj went on the attack from the outset.
Bastos again shot wide and Galatasaray went down the other end, with Umut Bulut shooting over.
The hosts almost took the lead after 12 minutes with Hamit Altintop's 25-yard shot, which Mario Felgueiras did well to repel in the Cluj goal.
Seven minutes later Cluj were ahead, Nounkeu putting into his own net from 12 yards.
Altintop again tested Felgueiras as the hosts looked to respond, with Nordin Amrabat shooting over.
And 10 minutes before the break they were handed a golden opportunity to equalise when they were awarded a penalty for a Laszlo Sepsi handball. Melo fired a fierce shot towards the bottom left of the goal but Felgueiras guessed the right way and parried the ball away.
Selcuk Inan and Albert Riera both had long-range efforts for the hosts in the opening minutes of the second half.
Emmanuel Eboue shot from distance on the hour and Sabri Sarioglu and Melo also sent in efforts, but none seriously threatened the Cluj keeper.
However, all their dominance paid off in the 77th minute when Yilmaz equalised, heading Amrabat's cross into the centre of the goal from close range.
Cluj had not shown as an attacking force at all in the second half and Galatasaray continued to go forward looking for the winner.
Selcuk Inan fired a free-kick over the bar and in the final minute another Inan set-piece found Yilmaz in the six-yard box, but he could only direct his shot wide.