The Croatian completed his move from Rangers on Tuesday night for an undisclosed fee, signing a four-and-a-half-year deal, and was introduced to the Goodison Park crowd at half-time during their 1-0 win over Barclays Premier League leaders Manchester City.
Darron Gibson's goal was only the Toffees' 23rd in 23 league games this season, and there will be an extra weight on Jelavic's shoulders following the departure of Louis Saha to Tottenham.
Moyes said: "It is great news for us. We have been looking for a centre-forward for a while. We just hope now he can help us score some goals and we can build on the performance (against City).
"Denis (Stracqualursi) has done great and with Jelavic coming in hopefully it bodes well for the coming months. I think he would have seen the hard work that the players put in.
"He is coming from a great football club but I think he is coming to another great football club. He really wants to play in the Premier League and he really wants to play for Everton. That goes a long way.
"He will also want to play for Croatia in the European Championship so hopefully that will keep his form good, keep him hungry to score and play well."
Jelavic scored 36 times in only 55 appearances for Rangers having been signed for £4million in the summer of 2010 from Rapid Vienna.
The 26-year-old knows he is making a step up in terms of quality but hopes to be able to maintain the standards he has set for himself.
"It is a big thing for me and it is a new step in my career, a step forward," he told evertontv.
"I am very pleased and very excited, it is a new challenge for me.
"The Scottish Premier League is not as high a level as here in England but I think it will help a lot as it is not so much a big difference.
"I think I can bring some good football and to score many goals and to try to win some trophies."
Jelavic was Moyes' third signing of the January transfer window, following a loan for Los Angeles Galaxy forward Landon Donovan and a permanent deal for Gibson from Manchester United, and his arrival was swiftly followed by the return of midfielder Steven Pienaar on loan from Tottenham.
The South Africa international left the Toffees a year ago but has struggled to make an impact at White Hart Lane and will spend the rest of the season in Liverpool.
Both clubs worked hard up until the deadline at 11pm to ensure the deal went through and Pienaar, who made more than 100 appearances for Everton in his first spell, said on Twitter: "Happy days. Look like we just beat the clock. Holding thumbs for next few hours."