The Toffees outplayed Scunthorpe United in Saturday's Cup triumph.
Osman, who stood in as captain for Neville, said: "It is not just the manager who is desperate - throughout the club we want to step and try to get some silverware now.
"This is a really good opportunity. Everybody needs some luck with the draws but we need to put on performances like this. Fingers crossed, we can go far.
"It was a good win, five different goalscorers, and hopefully it breeds confidence going into the derby next week.
"It was always going to be difficult and it was important we got a good start.
"We got out of the blocks fast and got 2-0 up at half-time.
"But we still knew the tie wasn't over and they proved that straight after half-time with a good goal."