Callum Paterson scored a quick brace to set up the win before a second-half goal from Jordon Forster gave Hibs some hope, but they were unable to level in the last Scottish Premiership clash between the teams for at least 16 months.
The first goal arrived when Paterson (37) climbed above Forster to meet a corner from the left and his downward header bounced over Liam Craig into the side of the net.
And four minutes later Paterson powered another header into the same corner of the net when he connected with Kevin McHattie's free-kick from the opposite side of the pitch.
Hibernian were given a lifeline on 69 minutes as Hearts struggled to deal with a quick free-kick and Ryan McGivern swept his cross to the back post for Foster to bullet his header in off the frame.
Following a hectic opening 20 minutes Hibs created the first opportunity when a glancing header from Jason Cummings drifted just wide of the target from a Craig delivery.
Sam Stanton was a stride short of latching on to a threaded pass into the Hearts penalty area from Craig, and a minute later Jamie MacDonald scrambled across his six-yard box to save a powerful drive from Cummings shortly before the opening goal.
Alex Harris' shot was deflected into the side netting in the opening exchanges of the second half as Hibs attempted to get back into the game.
At the other end Ben Williams had to be on his toes to deal with a 30-yard drive from McHattie, which the keeper did well to clutch.
Substitute James Collins had an opportunity to level the match five minutes from time when the ball spun to him inside the box but the Hibs striker swiped at fresh air with keeper MacDonald exposed.
Ryan Stevenson almost added a third for the visitors on 90 minutes when he beat the offside trap and lifted the ball over the advancing Williams, only to see it bobble wide of the far post.