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 advanced to the next stage, but the Reds needed extra-time to overcome Blackpool.
It did appear that it would be a straightforward contest for the Premier League outfit when Danilo added to an opener from Andrew Omobamidele to send them 2-0 ahead early in the second period.
Blackpool had other ideas, though, halving the deficit in the 61st minute through Albie Morgan, before Kyle Joseph levelled the scores in the 78th minute, and extra time was needed in the third-round replay.
The game's decisive goal came from Chris Wood in the 110th minute, with the striker having the simplest of tasks to tap a low Ryan Yates cross into the back of the net from close range.
Forest will take on Bristol City in the fourth round of the competition, with the Championship side impressively eliminating West Ham United on Tuesday night.