The Merseysiders came from behind to win at Goodison Park with headers from Denis Stracqualursi and Marouane Fellaini sealing the home side's progress following Danny Murphy's controversial penalty opener.
John Heitinga was penalised in the 13th minute when the ball struck his arm as he attempted to block a Damien Duff shot but Everton otherwise dictated the game.
Round said: "I think we deserved it. Their goal was against the run of play. It was a very difficult decision for the referee to give and I don't envy him in those positions.
"It did hit his hand but I think it was more ball to hand than hand to ball. We didn't think it was a penalty, it was quite harsh, but to come back, the team showed great spirit, great togetherness.
"We got the goal and then I thought we would go on and win it.
"We showed more ambition and I thought every time we put the ball in the net from a shot or cross we looked a real threat."
Stracqualursi's goal was his first for the club in only his third start.
The Argentinian has struggled to make an impact since joining on a season long loan from Tigre but justified his selection ahead of Louis Saha with his 27th-minute equaliser.
Round said: "I am really pleased for him, he is a terrific lad. He comes in every day to the training ground and works so hard. When you come from abroad it is difficult to settle into the Premier League but the lads have taken to him because of his work ethic.
"You saw that today - he ran and ran until he couldn't run any more. He got his just rewards, it was a great header. Hopefully he will get a few more."
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."
Everton chairman Bill Kenwright appeared to be harangued by a supporter after the final whistle but the club declined to comment on the incident.
Kenwright has been the subject of protests from a section of supporters frustrated at what they perceive to be stagnation under his ownership.
