Having enjoyed a silverware-laden decade through the 1980s, the Toffees' last triumph came in the 1996 FA Cup.
And despite punching above their weight in the Barclays Premier League - certainly in terms of the sums manager David Moyes have spent compared to their rivals - Round says ultimately only winning something tangible matters.
"I think a club like Everton needs to be up there winning trophies, it's in our history and tradition," said Round ahead of their Capital One Cup third-round tie at Leeds tonight.
"The manager is really focused on it. I am sure it gnaws at him every single day because that is what he aspires to, he needs to win trophies for Everton.
"Sustained success in the Premier League for Everton on the budget they have is definitely evidence of success.
"To be challenging up around European places year in year out in those circumstances could be deemed as being a successfully-run club.
"But obviously as staff and players you want to win a trophy.
"When you finish your career and you look back and see a medal in your locker it is very important."
Everton reached the FA Cup semi-final last season only to lose to city rivals Liverpool, who had already won the League Cup two months earlier.
But Round sees no reason, particularly after the start they have had in the league with three wins and a draw from five matches, why they cannot go back to Wembley and win this time around.
"This is a good opportunity because it provides a route into Europe," he added.
"It might not get the credence it deserves early on but when you get to the quarters and the semis you are certainly glad you are in.
"That is the outlook of the manager. He is going to be totally motivated to do his best and try to win this trophy if he can."
In terms of being a long way from success tonight's opponents Leeds are even further adrift, with their fall from grace well documented over the last few years.
But Round is wary that such an historic club, even one in the npower Championship, will always pose a tough test.
"It is a massive fixture. They are a really big club, Leeds," he said.
"I was explaining to one of our young players who has come over from Portugal how a big club they are and what it means to the fans of Leeds and what it meant to English football when they were at their pomp.
"Everton v Leeds is traditionally a massive fixture. There will be a big atmosphere and it is the sort of game I wish was still in the Premier League.
"They fell from grace very sharply and that was unfortunate but we have watched Leeds quite a bit and they will be no pushover as they are a decent team at home."