Coming into the game 22 points adrift of 11th-placed St Mirren in the Scottish Premiership with eight games remaining, Gary Locke's men knew defeat against their neighbours would see them consigned to the Championship.
However, goals from Dale Carrick and Billy King did enough to restore a measure of pride around Tynecastle against the 10 men of Hibs, who saw Alan Maybury sent off a minute from the end for a second bookable offence.
Hibs will feel aggrieved not to have earned a point, after Jordon Foster headed home with 10 minutes to go with the score still at 1-0, only for the referees' assistant to flag for offside when replays showed Foster to have been played on by a couple of yards.
The hosts took an early lead through Carrick, who lashed the ball home after a knock-down from Ryan Stevenson to send the home fans into raptures.
Hibs pressed hard in the second half and ought to have been level when Foster's goal was wrongly ruled out.
To rub salt into the wound, Maybury was then shown a second yellow for scything down King, who put the icing on the cake in stoppage time by rounding Ben Williams and slotting home.
The result earns Hearts a reprieve of sorts, although relegation will be confirmed if they lose at home to second-placed Aberdeen on Wednesday.
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