It was a far from convincing performance, which included conceding within a minute, but they just about managed to get the job done.
It is unlikely to ease the pressure on Rafa Benitez given this was a side 19th in the second tier. But a win is a win, and we have not had many of those recently.
You had to laugh when Everton conceded after 43 seconds.
We all feared the worst when Hull won a free-kick near the touchline, and right on cue the ball swung in found Tyler Smith completely unmarked to head the Tigers in front.
That kicked off a horrendous opening 15 minutes where Hull could have easily scored more.
Tom Eaves saw his header well tipped over by Asmir Begovic before the Toffees goalkeeper did well to push another effort from Eaves behind for a corner.
There have been few shafts of light in this grim season, but Demarai Gray and Anthony Gordon have been behind most of the rare bright moments - and they did so again here to turn the game around.
After controlling a cross-field pass Gray neatly exchanged passes with Gordon before sliding a first-time into the bottom corner for a well-taken leveller.
Gordon hit the post after surging run down the right as Everton finally began to exert some pressure. He then played a key role in Everton’s second, feeding Jonjoe Kenny down the right and his teasing cross was headed home by a diving Andre Gomes.
Gordon stung the palms of Baxter again in the second half as Everton pushed for a third. The scouser, along with Gray, Gomes and Godfrey were impressing - young players who should be central to any Toffees rebuild.
But the longer Everton failed to get a killer third the more you sensed an equaliser would come at the other end, especially when Benitez decided to bring Gordon off.
And it duly did.
Hull had offered little in the second half but this was a moment of quality as Ryan Longman curled a first-time effort from outside the area into the top corner.
There was no VAR at the MKM Stadium, it was bizarrely only in place in ties at Premier League grounds. Had it been in place here then Everton could have had two penalties, Jacob Greaves twice handling the ball in the box in each half but nothing was given.
Kevin Friend also bizarrely refused to give a foul when Rondon was dragged down on the edge of the box as he looked to get in on goal.
Don’t get me wrong Everton were poor, but they were not helped by some awful officiating.
Hull were inches away from taking the lead late on when Lewis Potter fired in a low effort from the edge of the box that smacked off the post.
Into extra-time we went.
The decisive moment came eight minutes into the restart, with Townsend sending in a swerving drive that deceived Baxter and flew into the net from 30 yards.
Gray was spent by this point so Benitez finally granted Evertonian wishes by bringing on Lewis Dobbin at the start of the second half of extra-time.
Eaves thought he had levelled in the second period but his volley from eight yards was superbly blocked by Begovic, who showed athleticism that belied his 34 years.
Cenk Tosun even made an appearance late on and saw an effort well blocked by Baxter, as the Toffees just about saw the game out.