The home side might have taken the lead after six minutes when Kris Commons advanced on goal, but his effort was pushed wide by Jamie MacDonald, before James Forrest fired a chance wide moments later.
The opening period was somewhat marred by an injury picked up by MacDonald following a clash of heads inside the penalty area, with former Aston Villa stopper Peter Enckelman joining the play in the early stages.
It was the home side that continued to dominate possession in the first half, but Hearts passed up a fine opportunity to take the lead when Callum Paterson struck wide from inside the box.
Celtic again came close to taking the lead just before the 30-minute mark, but Georgios Samaras struck the post from close range. However, the Greek striker had his goal less than five minutes later when he converted from close range to secure a half-time lead.