The visitors often looked like they had the potential to cause problems for Celtic when they did get forward, and they created the first chance of the game for Constantin Budescu, who struck a powerful effort just wide.
Despite seeing plenty of the ball, Celtic struggled to enjoy much success in the final third of the pitch as the stubborn Giurgiu defence limited the space on offer.
Craig Gordon made a vital save to keep out a low strike from the lively Budescu midway through the first half to ensure Celtic remained on level terms.
Celtic improved as the first half progressed, but they still looked vulnerable at the back, with Gabriel Enache dragging a great chance wide of the post.
Stefan Scepovic should have handed the hosts the advantage before the break after finding space inside the box, but he could only head over from close range.