Aaron Hickey scored his first Hearts goal in the 84th minute to lift his side off the bottom of the Ladbrokes Premiership and seal their first league win in 13 matches.
Levein had been backed by Hearts owner and fellow board member Ann Budge in midweek but saw Stevie Mallan's superb long-range strike put Hibernian ahead two minutes into the second half.
However, Uche Ikpeazu levelled 20 minutes from time before Hickey found the winner and it is Hibs boss Paul Heckingbottom who is now feeling the pressure after his side dropped to 11th amid a run of one victory in 11 league matches.
Heckingbottom made a big call by dropping Ofir Marciano and handing on-loan Preston goalkeeper Chris Maxwell a league debut, while Levein made four changes.
Goalkeeper Joel Pereira and right-back Michael Smith returned from injury, the latter lining up on the left of a back three, but there was no Steven Naismith as the influential forward battles a hamstring problem.
The first half was riddled with errors and nerves, with neither side able to impose themselves on the game.
Ryotaro Meshino was making his first start for Hearts and showed some good touches, but the closest his side came in the opening period was a Sean Clare header that looped onto the top of the net.
Mallan lifted the game out of its stupor two minutes after the restart. Josh Vela won a second ball after a long kick into the Hearts half and Mallan had only one thought on his mind when he assumed possession more than 30 yards out. The midfielder struck a swerving shot into the roof of the net.
Meshino got the visitors' first strike on target as they sought an instant response, but Maxwell easily saved.
There was rare consensus among the rival supporters when Hearts fans joined in as the home crowd started singing 'You're getting sacked in the morning' in the direction of Levein.
The Hearts manager's predicament almost intensified when the ball fell for Melker Hallberg 12 yards out, but the Swede sliced a wild effort wide.
However, Hearts capitalised on a corner when Christophe Berra's header fell for Ikpeazu. The powerful striker held off several Hibs players, turned on the edge of the six-yard box and hooked a shot over the line.
Levein had turned to youth for salvation before the goal, bringing on Callumn Morrison and Andy Irving, and it was an even younger player who came to his aid.
Hickey received the ball wide on the left from Ikpeazu before cutting in and the 17-year-old fired a shot that took a slight deflection and looped over Maxwell's head into the far top corner.
It was the Hearts fans' turn to taunt Heckingbottom about his job security, with only a handful of Hibs fans joining in this time.
Hibs had a series of chances to equalise. Pereira made saves from substitute Glenn Middleton, Scott Allan and Ryan Porteous, while Jason Naismith missed a close-range header.
Substitute Christian Doidge had the ball in the net after heading it out of Pereira's hands, but there were few complaints when referee John Beaton blew for a foul and Levein soon joined in the celebrations in front of the jubilant Hearts fans.