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Andros Townsend struck a superb extra-time winner as Everton saw off dogged Hull in the FA Cup to ease the pressure on under-fire boss Rafael Benitez.
The Toffees suffered a nightmare start as Hull's Tyler Smith headed in George Honeyman's free-kick inside 45 seconds.
But Demarai Gray steered Everton level and Andre Gomes nodded them ahead.
Ryan Longman curled in a brilliant equaliser for the Tigers but Townsend's fierce 25-yarder deceived Hull keeper Nathan Baxter to put Everton through.
Hull, who hit the woodwork through Keane Lewis-Potter at 2-2 in normal time, came close to forcing penalties but Toffees keeper Asmir Begovic made a stunning save to deny Tom Eaves.
Victory was welcome cup cheer for Everton, who have exited the competition at the third-round stage in three of the previous five campaigns.
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Boost for Benitez - but it's far from convincing
When Everton fell behind less than a minute in and some travelling Toffees supporters joined in with the underdogs' time-honoured "you're getting sacked in the morning" chant, things appeared bleak for Benitez.
Coming in to the game on a run of just one win in 12 Premier League matches that has seen the Toffees slip to 15th after a decent start to his reign, the former Liverpool boss is still struggling for acceptance from the Goodison faithful.
His cause was not helped by Smith heading home his first Hull goal since a summer move from Sheffield United as Everton began poorly and were fortunate not to fall further adrift.
Begovic was twice called on to thwart tall striker Eaves, the first from a powerful header and the second a shot from a tight angle, and the visitors looked all at sea at set-pieces.
But then Anthony Gordon and Gray wove their magic, a classy exchange of passes in the Hull area culminating in the former Leicester man calmly tucking beneath Baxter.
Gordon has been one of Benitez's bright sparks recently and he was at the centre of everything good Everton produced in the first half, clipping the outside of the post after a mazy run and setting Michael Keane clear for a shot superbly tipped aside by Baxter.
The 20-year-old played a role in Everton's second too, teeing up Jonjoe Kenny for a cross that Gomes, timing his run into the six-yard area perfectly, dived in to head home.
It was a blow that appeared to rock Hull and the game looked to be meandering towards a routine Everton win when Longman found Begovic's top corner from 20 yards just seven minutes after coming on.
Everton rallied with Gray almost replicating Longman's strike, curling inches wide, and Hull's Jacob Greaves fortunate there is no video assistant referee at non-Premier League grounds in the FA Cup when his raised hand blocked a Gray shot.
The Toffees had their own escape at the other end when Lewis-Potter's low shot came back off Begovic's near post, but the Championship side had no answer when Townsend produced a trademark swerving, dipping strike from distance to decide it.
Line-ups
Hull
Formation 3-5-2
- 13Baxter
- 24Bernard
- 17McLoughlin
- 4Greaves
- 19WilliamsSubstituted forLongmanat 63'minutes
- 10Honeyman
- 6SmallwoodSubstituted forHuddlestoneat 63'minutesBooked at 90mins
- 8DochertySubstituted forMoncurat 63'minutes
- 11Lewis-Potter
- 22SmithSubstituted forHindsat 105'minutes
- 9Eaves
Substitutes
- 1Ingram
- 14Cannon
- 16Longman
- 18Moncur
- 20Smith
- 21Fleming
- 23Huddlestone
- 39Hinds
- 41Mills
Everton
Formation 3-4-3
- 15Begovic
- 23Coleman
- 5Keane
- 22GodfreyBooked at 114mins
- 2KennySubstituted forDoucouréat 74'minutes
- 6Allan
- 21André GomesSubstituted forGbaminat 88'minutes
- 19MykolenkoBooked at 45mins
- 24GordonSubstituted forTownsendat 66'minutes
- 33RondónSubstituted forTosunat 117'minutes
- 11GraySubstituted forDobbinat 105'minutes
Substitutes
- 1Pickford
- 3Patterson
- 4Holgate
- 14Townsend
- 16Doucouré
- 20Tosun
- 25Gbamin
- 61Dobbin
- 62Onyango
Referee:Andy Madley
Attendance:16,282
Live Text
Match ends, Hull City 2, Everton 3.
Second Half Extra Time ends, Hull City 2, Everton 3.
Corner, Everton. Conceded by Nathan Baxter.
Attempt saved. Cenk Tosun (Everton) left footed shot from the left side of the box is saved in the bottom left corner.
Attempt blocked. Allan (Everton) right footed shot from outside the box is blocked. Assisted by Vitalii Mykolenko.
Josh Hinds (Hull City) wins a free kick in the attacking half.
Foul by Vitalii Mykolenko (Everton).
Attempt missed. Jean-Philippe Gbamin (Everton) right footed shot from the centre of the box is too high. Assisted by Allan following a set piece situation.
Substitution, Everton. Cenk Tosun replaces Salomón Rondón.
Foul by George Honeyman (Hull City).
Andros Townsend (Everton) wins a free kick in the attacking half.
Tom Huddlestone (Hull City) wins a free kick in the defensive half.
Foul by Salomón Rondón (Everton).
Ben Godfrey (Everton) is shown the yellow card.
Corner, Everton. Conceded by Di'Shon Bernard.
Foul by Josh Hinds (Hull City).
Allan (Everton) wins a free kick in the defensive half.
Attempt missed. Josh Hinds (Hull City) right footed shot from very close range is close, but misses to the right following a corner.
Attempt saved. Tom Eaves (Hull City) left footed shot from the centre of the box is saved in the bottom right corner.
Corner, Hull City. Conceded by Séamus Coleman.
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