Fulham ended a miserable run of six consecutive defeats in the Premier League, as Rene Meulensteen earned his first three points as head coach since relacing Martin Jol at Craven Cottage.
The London outfit will now look to build on the momentum garnered by the victory, against Roberto Martinez's side on Saturday.
History suggests Everton will stretch their eight-match unbeaten run in the top flight, with Fulham having never beaten Everton at Goodison Park.
But Stekelenburg is confident Fulham can break their hoodoo so long as the side play to their full potential.
"Sometimes things change and we're making good strides now," the Netherlands international told the club's official website.
"If we keep playing like this, keep working hard in training and really get into a system of how we want to play, then we'll be a match for anyone.
"Everton will be tough but every game in this league is difficult and I'm confident we can get a result if we play like we did against Aston Villa."
Stekelenburg was particularly happy to claim a clean sheet against Paul Lambert's charges and was impressed with Fulham's defensive quality.
He added: "I'm very pleased with a clean sheet because we defended really well. But we have to play like this every game. It was a terrific day for us and we played well against a good opponent."
