An error from Hull goalkeeper Nathan Baxter ultimately sent the Toffees to the fourth round when an Andros Townsend strike flew down the middle of the goal in the first half of extra-time.
Benitez had barely sat down before Hull were ahead, when Tyler Smith headed home a George Honeyman cross after just 43 seconds, leaving the Spaniard livid.
Chants rang out from the home support over Benitez's impending doom, and BBC commentator Jonathan Pearce confirmed the feeling was felt all around the MKM Stadium
"'You're getting sacked in the morning' is the cry from Hull fans, and there are Everton fans, I can tell you, applauding that song," Pearce said.
It took 20 minutes for Everton got back in the game against the lower-league opposition, with impressive youngster Anthony Gordon exchanging passes with Demarai Gray who tucked in.
Andre Gomes gave Everton a half-time lead when he headed in a Jonjoe Kenny cross before the break, but Benitez's side failed to capitalise on their momentum after the break.
Hull continued to grow into the second half, before substitutes George Moncur and Ryan Longman combined to see the latter send an unstoppable curling shot past a helpless Asmir Begovic.
The Championship side kept the impetus to earn extra time, but it wasn't long before talkSPORT regular Andros Townsend ended their chances.
A swerving long-range strike in the ninth minute of extra time should have been handled by goalkeeper Nathan Baxter, who perhaps unknowingly helped Benitez survive another day.
Begovic too was key, making sure the Toffees kept their lead with a stunning point-blank save on a Tom Eaves strike that looked destined for the back of the net.
But as the game wore down, the banners came out, with Everton fans making their feelings towards their unwanted manager loud and clear.
'Benitez get out of our club' read the highly visible message, and you get the feeling that it might not be long before the Everton fans get their wish.
Elsewhere in the late kick-offs, Chelsea steamrolled non-league Chesterfield 5-1 while Southampton needed a Shane Long extra-time goal to down Swansea 3-2.
An Emiliano Marcondes hat-trick saw Bournemouth beat National League side Yeovil 3-1 and a loan goal from Ryan Law saw Preston beat Birmingham from the league above