Hearts knew defeat would confirm their Scottish Premiership fate but Carrick's early strike looked enough to earn Gary Locke's side their third derby win of the campaign before King sealed the points in injury-time, moments after former Hearts defender Alan Maybury had been sent off.
Hibs fans came to Tynecastle in party mood with numerous green balloons occupying the goalmouth in front of the Roseburn stand as they prepared to celebrate their rivals' relegation.
But they will have left more than a touch concerned about their own top-flight status after their team failed to seriously trouble goalkeeper Jamie MacDonald, although Jordon Forster had a goal controversially disallowed during a sustained spell of late pressure.
Hibs finish the weekend just six points above St Mirren, who occupy the relegation play-off place, with six games to play.
They have now recorded just one win in 13 games since their New Year derby win and Terry Butcher's bid to stop the rot saw him make six changes to his side, with captain Liam Craig among those dropping to the bench.
Hearts will go down on Wednesday if they lose to Aberdeen but they looked a team determined to avoid the inevitable blow being inflicted by their deadly rivals.
Hearts fans had already regaled their visitors with a chorus of "You're going down with the Jambos" when Carrick struck inside six minutes.
Jamie Hamill won possession after his free-kick was half-cleared and fed the ball to right-back Callum Paterson.
Ryan Stevenson knocked his deep cross down to Carrick, who had time and space to sweep the ball home from eight yards.
Hibs striker James Collins soon headed not far wide from Maybury's cross but Hearts were quickly back on the front foot and David Smith had a shot saved by Ben Williams.
Hibs took until the halfway point of the half to find any sort of rhythm to their attacking play but Danny Handling and Sam Stanton were both off target when giving shooting chances.
The visitors missed a great chance to equalise on the counter-attack five minutes before half-time. Duncan Watmore turned inside Kevin McHattie and laid the ball back for Collins but the striker leant back and shot over from 15 yards.
Handling shot wide from 22 yards after a decent move but Hearts finished the half the stronger side.
Stevenson broke and fed Carrick on the edge of the box but he turned and shot over when Smith was bursting forward outside him.
And Sam Nicholson hit a dipping 30-yard strike that Williams pushed away at full stretch.
Stanton wasted another in a series of free-kicks just after the restart and Hearts soon broke as Carrick set up Smith, but the winger fired an angled drive just wide from 20 yards.
McGivern headed just over from Stanton's near-post corner but again Hearts came closer moments later when Smith drove the ball across the face of goal.
Hibs brought on Alex Harris and Kevin Thomson as the game entered the final quarter and the former had a rare shot on target but it was too tame and far out to trouble MacDonald.
MacDonald saved a couple of efforts from Collins, both of which had the sting taken out of them by Hearts defenders.
Hibs then pushed Hearts deeper and their pressure looked to have told when Forster headed home Stevenson's cross. The linesman immediately raised his flag but replays showed he was onside.
Their comeback hopes were dashed in the 89th minute when Maybury received his second yellow card for bringing down Hearts substitute King, which sparked a mini pitch invasion from one visiting fan.
And King delivered the knockout blow when he took the ball round Williams and slotted home after being sent clear by Stevenson.