The Toffees, whose last silverware came in the competition in 1995, laboured to an extra-time 2-1 win over Championship strugglers Rotherham to progress to the fourth round for only the second time in five seasons.
Manager Carlo Ancelotti made eight changes for the tie but still fielded a side which contained half-a-dozen first-choice players, including the likes of James Rodriguez, Michael Keane, Andre Gomes and Seamus Coleman.
"It is very important, it is a priority for us, it gives us a chance to win a cup for the fans," said Ferguson, a member of the 1995 cup-winning side.
"It is a big competition for us and hopefully we can go as far as we can. We will carry on picking strong teams and take it really seriously. We won't be chopping and changing.
"We are taking it seriously and picking the strongest team we could pick and hopefully we can go as deep as we can."
Having taken an early lead through Cenk Tosun's first Everton goal since November 2019, Ancelotti's side dropped off the pace and Rotherham took control of the game and could have scored a couple before their deserved equaliser just after half-time through the excellent Matt Olosunde.
Paul Warne's side continued to match their Premier League opponents, with Olosunde giving France left-back Lucas Digne, returning from injury after six weeks out, a torrid time.
However, the quality of the Toffees' bench told as one of their substitutes Abdoulaye Doucoure scored early in the first extra period but it was far from a convincing victory.
"I thought there were some amazing performances and Matt Olosunde got the goal that his deserved, I thought he was a thorn in their side from start to finish," said Warne.
"He was good and I don't know if it was the end of the second half or extra time, but Carlo changed his shape to match us up in the middle because we were over-running them in there, which is a right compliment.
"It was a really good experience for my younger players and if I could have got everyone on, I would have. I wanted them all to get a feel of these big games and enjoy it."
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