Callum Paterson had the first chance after eight minutes, but the forward couldn't find room to get his shot away and the attack fizzled out.
The away side saw more of the ball in the opening 10 minutes, but couldn't work any clear-cut chances.
Jason Cummings had a chance for Hibs on 19 minutes, but his glancing header went just wide of the right post.
Cummings tested the goalkeeper again on the half-hour mark with a long-range strike, forcing a good save from him.
Hearts then took the lead on 37 minutes through Paterson, who headed home at the back post from a corner to make it 1-0.
Four minutes later, Paterson scored his second to make it 2-0, heading past the goalkeeper powerfully from a free kick.
Hibs started the second half strongly as they looked to get themselves back into the game.
The home side pulled a goal back on 68 minutes through Jordon Forster, who reacted quickest to a ball in the box and headed past the keeper.
Hearts had a huge let off on 84 minutes when a failed clearance rolled into the path of James Collins; however, the striker missed the golden opportunity to bring his side level.
Ryan Stevenson almost wrapped up the points for Hearts on 89 minutes after beating the offside trap, but the midfielder's chip over the goalkeeper trickled just wide of the post.