Fulham boss Martin Jol was frustrated that his team failed to gain a greater foothold in the contest but had no complaints about the result.
The sides were evenly matched on paper, with Fulham leading the Merseysiders in the Premier League only on goal difference, but Everton looked the more inventive.
Jol said: "If you score a goal that early, you hope for more, you hope to go to the next round. But you have to defend well because you know the quality they have from crosses and midfield players.
"They had a strong spell of five corners in about two minutes, that is their strength. They were trying to get balls into the box from the right and the left and we knew they would.
"Overall we defended well but on two occasions we didn't and there was another ball on to the bar which made us nervous, but with our quality we should have done better.
"Under pressure you have to play around them and we couldn't do that most of the time. The result is disappointing but overall I think Everton deserved it.''