The Croatian, who has scored 15 goals since moving from Rangers last season, said that the Blues have a "strong" team, capable of mixing it with Europe's elite.
The Merseyside club currently sit fourth in the Premier League thanks to a flying start, winning four of their eight games and losing just once.
"We are a strong team," he said. "I have heard all about the success that Everton enjoyed in the mid-'80s and what a great team they had back then. I can't promise that we are ready to get back to those levels just yet, I do believe that we are strong enough to play Champions League.
"We have already beaten Manchester United this season, and last season we beat Manchester City, so we have shown that we can play and also that we can compete."
Jelavic said that he is determined to score his first competitive win over local rivals Liverpool in Sunday's derby match.
"I have had two bad experiences playing against Liverpool in this country, but I did beat them in a pre-season friendly when I was with Rapid Vienna," he added. "Obviously, it is a big derby and it is a big game for everyone and now I want to win against Liverpool with Everton."
Jelavic has scored four goals this season.