The hosts went into Sunday's Edinburgh derby needing at least a point to fend off the chop for the time being, having kicked off 22 points adrift at the base of the table with just eight games remaining.
Hearts opened the scoring inside seven minutes when, from Callum Paterson's long cross into the box, Ryan Stevenson laid the ball into the path of Carrick who smashed it past Hibs keeper Ben Williams from eight yards.
And that was the way it remained until deep into second-half injury-time, when the hosts added another goal on the break as Stevenson fed King who rounded keeper Williams before slipping the ball into the back of an empty net.
By then Hibs had seen a Jordan Forster header ruled out for offside by referee Steven McLean - although TV replays suggested otherwise - with the visitors reduced to 10 men in the 89th minute when Alan Maybury collected a second card for a wild lunge on substitute King.
Hearts, however, just about deserved the win with Sam Nicholson almost adding a second goal during the final few seconds of the first half when his fierce strike from 30 yards was brilliantly saved by Williams who was fully-stretched to push the ball away for a corner.
David Smith flashed a low shot inches wide of the left post as Hearts went close again early in the second half, while Daniel Handling and Jim Collins looked dangerous at times for Hibs.
Yet, despite the win over their closest rivals, Hearts' status as a top-flight club appears doomed, with relegation likely to be confirmed in early April and possibly as early as this Wednesday when Aberdeen visit Tynecastle.
Hibs, meanwhile, have won just one of their last 13 games and are only six points above St Mirren who currently occupy the relegation play-off spot.