The Merseysiders came from behind to win 2-1 at Goodison Park with headers from Denis Stracqualursi and Marouane Fellaini after a controversial Danny Murphy penalty. 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."
He added: "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."
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.
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."
Source: PA
Source: PA