Both sides struggled to get a hold on the game early on, resulting in a scrappy opening 20 minutes.
Rowan Vine had the first effort on goal after 32 minutes, but his strike went straight at the goalkeeper.
Kevin McHattie hit a free kick just wide of the mark on the 42-minute mark as Hearts continued to pressure.
After the break, Jamie Hammill hit a strike from distance, but the goalkeeper just kept it out with a fine save.
Two minutes later, Scott Robertson came close with a diving header, though he couldn't find the target.
James Collins had a great chance for Hibs on 63 minutes, but his low shot from the left was just kept out.
The away side had another huge chance minutes later through Tom Taiwo, but the midfielder could only fire over the bar.
Hearts took the lead on 71 minutes when Callum Paterson met a cross from the right and fired a header into the roof of the net to make it 1-0.
Vine almost doubled the home side's lead on 79 minutes, but the forward could only send his shot wide of the post.
Hibs piled on the pressure in the last five minutes, but they couldn't find the equaliser.