The London club were knocked out at this stage in 2011 against Chelsea, before being dumped out a round earlier against Sheffield Wednesday last year.
They have never managed to progress beyond the quarter-finals in the competition's 53-year history.
Their poor record was nearly extended when they almost fell at the first hurdle this time around, with a Hugo Rodallega strike three minutes before the end of extra time taking their clash with Burton Albion to penalties - before they eventually ran out 5-4 winners.
There was a similarly uncomfortable passage into the third round for Roberto Martinez’s Everton, who needed a Marouane Fellaini goal in extra time to see off League One outfit Stevenage at Goodison Park.
This game is one of six all-Premier League ties in the draw and, with both sides not in Europe this term, they will surely view the competition as a potential route back to continental football.
Everton have never won the trophy, having lost out in the final in both 1977 and 1984. The first of those instances, against Aston Villa, was decided by a third replay as the first two clashes ended in draws, Villa eventually running out 3-2 winners.
In 1984, their initial final meeting with Merseyside rivals Liverpool ended in a 0-0 draw before the latter won the replay at Maine Road 1-0 thanks to a goal from Graeme Souness.
Victory at Craven Cottage on Tuesday would earn Everton the chance to progress beyond the fourth round of the competition, something they have only achieved once in the last 25 years.
That came in the 2007-08 season, when David Moyes' side reached the semi-final stage only to bow out 3-1 on aggregate to Chelsea.
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