But the victory did little to ease the pressure on manager Rafa Benitez.
Everton made a nightmare start and fell behind after just 45 seconds.
Hull were awarded a free-kick and Tyler Smith was left unmarked to head home.
But after a lacklustre first 20 minutes, the Blues bucked up their ideas and Demarai Gray equalised with a composed finish in the 21st minute.
Then 10 minutes later the visitors were head when Andre Gomes headed home inside the box.
The second period was a lot quieter and Everton had the better of things - until the Tigers clawed their way back into the game on 71 minutes.
Following a triple substitution from Grant McCann, George Moncur teed-up Ryan Longman, who wonderfully curled his shot into the top corner.
That took the game to extra-time and it was evident both teams were tiring.
But in the 99th minute, substitute Andros Townsend’s swerving effort found the back of the net and put Everton into the fourth round of the Cup.
- Andros Townsend: the winger showed his commitment by coming back from fitness early to make the bench. Townsend was then introduced to make a difference. Indeed, he did that by settling matters for Everton with another trademark long-range strike.
- Demarai Gray: the ex-Leicester wide man few poor decisions throughout and ran offside carelessly on a couple of occasions. But Gray was Everton’s chief attacking threat and always put the Hull defence on the back foot when he got the ball at his feet. His equalising goal was the moment of quality the away side were crying out for.
- Asmir Begovic: the veteran keeper had no chance with either of Hull’s goals. However, coming in for Jordan Pickford, Begovic made two big saves from Tom Eaves. His stop in the second half of extra-time when he got down low to his left was as good a save you’ll see all season.
-Salomon Rondon: the striker remains a boo boy among the Everton faithful. Struggled yet again when leading the line. There was one smart header to release Anthony Gordon but, in general, Rondon lacked any sort of threat.
- Rafa Benitez: this victory will do nothing to win over the Everton boss’ detractors. Everton toiled against a bottom-half Championship side and were second best for significant periods of the game.
- Kevin Friend: the referee did not give Everton a penalty on two occassions for handball. There was also a curious decision by not awarding Rondon a free-kick when on the edge of the box when he was dragged back by George Honeyman.
Hull: Baxter 5, Bernard 6, McLoughlin 6, Greaves 6, Williams 5 (Longman 63; 7), Smallwood 5 (Huddlestone 63; 6), Docherty 6 (Moncur 63; 6), Honeyman 7, Lewis-Potter 7, Eaves 6, Smith 6.
Everton: Begovic 7, Kenny 6 (Doucoure 74; 6), Coleman 6, Keane 6, Godfrey 6, Mykolenko 5, Allan 6, Gomes 6 (Gbamin 88; 6), Gordon 6 (Townsend 66; 7), Gray 7 (Dobbin 116), Rondon 4 (Tosun 117).
What’s Everton’s next fixture?
The Toffees are back in action on Tuesday evening.
They welcome Leicester City to Goodison Park for the rearranged fixture that was cancelled on 19 December. Kick-off is at 20.00.