The Swans suffered their first home defeat of the new Premier League season over the weekend with a 1-0 loss to high-flying Southampton.
Monk now wants his side to overcome Everton at the Liberty Stadium and put together a morale-boosting run in the Cup.
"We are at home, I am sure the fans will be right up for it and hopefully we can get the right result," he is quoted as saying by Sky Sports News.
"The Cup always ranks high and it would be nice to go on a run. We also want to bounce back because we lost to Southampton on Saturday and it would be nice to try to make sure we don't lose the next one."
Swansea are currently in fifth place in the Premier League table after taking nine points from their opening five games.