In extra time, Man-of-the-match Marc Roberts turned Gerard Deulofeu’s drilled cross into his own net and Romelu Lukaku finished off a flowing breakaway move as the Tykes bowed out of the competition.
The Reds led 2-0 at the break after Sam Winnall swept home a loose ball and Marley Watkins bundled home from close range. Kevin Mirallas halved the deficit six minutes into the second-half and Steven Naismith completed the turnaround just before the hour.
Dan Crowley superbly slotted the Reds back ahead one minute later only for Romelu Lukaku to scramble another equaliser with 12 minutes remaining.
The half began in frantic fashion with Josh Scowen firing inches wide in the opening minute.
The Toffees responded instantly as Bryan Oviedo cut inside but failed to hit the target. Adam Davies produced a stunning double save, first to deny Kevin Mirallas’ swerving drive and then Aiden McGeady’s follow-up from close range.
Not to be outdone, Tykes forward Sam Winnall connected firmly with Conor Hourihane’s cross but failed to find the net. On 16 minutes, Mirallas played a neat one-two with McGeady and let fly, only for Davies to again deny him while Romelu Lukaku was unable to connect with Matthew Pennington’s dangerous delivery.
Twenty-two minutes in and the Reds had seized the initiative. Hourihane fought off both John Stones and Matthew Pennington, the ball broke for kindly Sam Winnall and the striker found the bottom corner in clinical fashion.
Six minutes later and the Reds doubled their lead through Marley Watkins. Marc Roberts crossed to back post, Josh Scowen headed back across goal and Watkins bundled the ball home from close range.
Winnall was cautioned for tripping Muhamed Besic on 33 minutes and a moments later, Roberts put his body on the line to deny Lukaku from eight yards out.
McGeady tested Davies with a curling left-footed drive from the edge of the area on 36 minutes while Mirallas fired over after a jinking run. Roberts and Wabara prevented any further damage with a series of tremendous blocks, ensuring Barnsley ended the half with a two goal cushion.
Half time substitute Ross Barkley could have scored seconds into the second-half only for Josh Scowen block before Barkley blazed over.
Scowen was unfortunate to fire over the bar from 25 yards on 50 minutes but Everton halved the deficit one minute later as Mirallas acrobatically netted from inside the area.
Davies superbly denied Gerard Deulofeu with his feet but could do nothing to stop Steven Naismith equalising on 59 minutes with a header. The Tykes regained the lead a minute later as Reece Wabara broke clear down the line and pulled back for Dan Crowley who elegantly slotted home his first goal for the club. Deulofeu pulled back for Lukaku on 62 minutes but the Belgian’s scuffed effort was hacked away on the line by the covering Wabara. Hourihane flashed a free kick centimetres wide on 63 minutes while the lively Deulofeu blazed over after riding three Reds challenges.
Ben Pearson was cautioned for dissent but the Toffees drew level again 12 minutes from time as Deulofeu crossed for Lukaku to make it 3-3 from close range.
With 11 minutes remaining, Barkley volleyed just wide from the edge of the box and Naismith went close to grabbing a winner three minutes later as Lukaku linked with Naismith but the Scot’s effort drifted past the post.
Wabara threw himself in front of Lukaku’s goal-bound effort and Roberts prevented Phil Jagielka grabbing the winner with a towering header.
On 89 minutes, Mirallas rose highest to meet Deulofeu’s teasing cross but failed to direct his header on target as the game ended 3-3 after 90 minutes, forcing extra time.
Alfie Mawson headed Conor Hourihane’s teasing delivery wide two minutes into extra time and substitute Conor Wilkinson blasted straight at Joel Robles. Six minutes into extra time, Deulofeu worked himself a yard of space and his drilled crossed was diverted home by the unfortunate Marc Roberts, giving Everton the lead for the first time on the night.
Mirallas could not add to his tally after lifting Tom Cleverley’s through ball over the crossbar while Tykes substitute George Smith fizzed a low ball in that no one could connect with.
Barnsley threw the kitchen sink at the Toffees in the second period of extra time as Josh Scowen worked some space but could not beat the towering frame of Robles with a powerful drive. Conor Hourihane struck the crossbar on 107 minutes with a fantastic effort while man-of-the-match Roberts received a caution for tripping Steven Naismith moments later.
With 114 minutes on the watch, Davies denied a Romelu Lukaku rocket and thwarted Ross Barkley’s curling effort. Five minutes from time, Barkley broke clear and fed Lukaku who ensured it was to be Everton who faced Reading in the third round by beating Davies at his near post.
A crowd of 12,290 witnessed the thriller with 4,215 making the trip from Merseyside to see their team progress 5-3 after extra time.
Reds: Davies, Wabara (Rothwell - 94' ), Roberts, Mawson, Nyatanga, Scowen, Pearson, Watkins (Smith - 105' ), Crowley, Hourihane, Winnall (Wilkinson - 83' )
Subs Not Used: Townsend, Digby, Maris, Templeton
Everton: Robles, Pennington (Barkley - 45' ), Stones, Jagielka, Oviedo, Besic, Cleverley, McGeady (Deulofeu - 45' ), Naismith, Mirallas (McCarthy - 110' ), Lukaku
Subs Not Used: Kone, Howard, Browning, Holgate