The hosts took the lead via a Leighton Baines penalty on 20 minutes but Wilfried Bony drew the Swans level by the break.
But two goals in five second-half minutes from Romelu Lukaku and Ross Barkley gave the Toffees a cushion Monk's side could not overhaul.
While Ashley Williams pulled a goal back in injury time, the Swans could not score from other chances they created as they slumped to another defeat to stay four points above the drop zone ahead of a trip to Arsenal in midweek and next Saturday's clash with fellow relegation battlers Norwich.
Monk agreed afterwards that his side could have taken something from the game, telling Sky Sports: "We knew coming here, they'd be periods where Everton would be strong and they'd have a sustained period of pressure.
"I said to the boys in the changing room there, it's about being able to deal with that side of it and those minutes that you are suffering and come through it unscathed, have the right mindset to deal with it.
"And unfortunately in those periods we didn't do well enough to repel their attacks and come through it unscathed because you could see, once you come through those periods, there's plenty of chances to be had and we had plenty of chances.
"So I think the point for the boys there, what I said to them there, was getting through that period of pressure and make sure we manage that period of pressure properly and, like you said, you can push on and get goals."
Monk continues to focus on trying to pick up points as quickly as possible to make the Swans safe.
"We go again - we've got a game Tuesday and we've got a very important game again on the weekend and they're all important games, including this one we've just had.
"It was a chance where we didn't take any points so we need points as quickly as possible, so we have to make sure we try and take some from the next game.
"We'll be working hard on the training ground and trying to get what we need (from the remaining games) to make us safe for the season."