Goals from Nathan Dyer, Gylfi Sigurdsson and Marvin Emnes ensured that the Swans moved a step closer to repeating their triumph in 2012, and Monk has revealed that he considers the competition as an important part of their season.
The 35-year-old is quoted by BBC Sport as saying: "It was very professional. We played a very good team in Everton. We started well and created quite a few good chances. We took the goals very well.
"Everyone knows how we've done in the cup in recent history. It's an important competition to the fans."
Swansea have also started their Premier League campaign in fine style, registering nine points from their opening five fixtures.