The Welsh side were on course to collect a point at Everton when Wilfried Bony cancelled out Leighton Baines' first-half penalty just after the half-hour mark, but they went on to lose 3-2 after conceding twice in the opening 15 minutes of the second period.
Romelu Lukaku and Ross Barkley netted for the hosts to put the game out of Swansea's reach, despite Ashley Williams's stoppage-time header.
The defeat sees Swansea remain four points clear of the bottom three in the Premier League but head coach Monk, who has not overseen a victory in any competition since his side's 3-0 win over rivals Cardiff City in his first game in charge last month, is confident he can preserve their top-flight status.
"If I thought I wasn't doing the right things then I'd hold my hands up," he said. "If you asked the players who see how hard I work, I think they'd say I'm doing the right things.
"I'll be judged on my results but until I'm told differently I'll continue to work as hard as possible."
Monk blamed lapses of concentration at the start of the second half, which allowed Everton to establish a two-goal cushion in a five-minute spell before the hour from which the visitors could not recover.
"When you come to places like this, there's always a period where the opposition has some pressure, and you have to manage that properly," he added.
"You have to get through those periods unscathed, you have to dig in, make the right decisions and we didn't do that in a five-minute period in the second half and it cost us.
"And it gave us too much of a mountain to climb.
"After that we had a few chances and if we hadn't given them such a lead we would have been able to come away with something."
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