Andre Gomes netted the only goal of the all-Premier League contest between Everton and Crystal Palace at Goodison Park, with his 42nd-minute effort sending Sean Dyche's side into round four.
The 30-year-old was an unused substitute in the goalless draw with Aston Villa in the Premier League on Saturday, but he showed his class with a brilliant free kick to decide Wednesday's contest.
Both managers named strong XIs, and it was an even match, with both sides having 11 attempts, five of which were on target, but Gomes's set piece was ultimately the difference.
Everton's reward for their progression is another all-Premier League contest, with the Merseyside outfit set to welcome Luton Town to Goodison Park in the fourth round on January 27.
Norwich also got the job done in normal time on Wednesday, advancing to the fourth round of the competition courtesy of a 3-1 victory over Bristol Rovers.
The home side took the lead in the 20th minute of the contest through Luke McCormick, and the League One outfit led 1-0 at the half-time interval.
Norwich stepped it up in the second period, though, levelling the scores in the 53rd minute through Gabriel Sara, before Adam Idah turned the match around from the penalty spot in the 59th minute.
Bristol Rovers were pushing for a leveller which would have forced extra-time, but the game's fourth goal came for the Championship side in the 87th minute, with Kenny McLean on the scoresheet.
Norwich have therefore progressed to the fourth round, with the Canaries being rewarded with a trip to Anfield to take on Liverpool.
Forest have also
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