Jelavic has made a significant impact at his new club since joining from Rangers in January, netting 11 times for the Merseysiders in just 16 appearances
His latest strike came in yesterday's [Sunday] 3-1 home victory over Newcastle, after which Moyes expressed his delight at the way the 26-year-old Croatian has fulfilled a role the Everton team had been crying out for.
The manager was keen to stress, though, that having had years of consistency from some of his other key recruits during his decade-long tenure with the Toffees, he wants to see no let-up from Jelavic.
"He has been a great signing for us, but my real signings are the people like Tim Cahill, Phil Neville, Tim Howard - the ones who have been here for six, seven or eight years and consistently done it," Moyes said.
"Niki has been here for a few months - what we are saying is that we want to see him doing that year in, year out like these boys have done.
"But he has certainly given us something to think about and the supporters something to cheer about.
"We have missed someone who can be a number nine at Everton really and for the moment Niki has done it well."
Jelavic finished smartly at the second attempt in yesterday's Barclays Premier League final-day clash after Newcastle goalkeeper Tim Krul had blocked his initial shot.
That 27th-minute effort followed Steven Pienaar's deflected opener for Everton (16) and John Heitinga's header (65) then made it 3-0, with Tony Hibbert's own-goal eight minutes later proving little more than a consolation for the Magpies.
Other results yesterday [Sunday] meant that even if the Tyneside outfit had won, their bid to secure a top-four finish would have been unsuccessful and their fans will certainly not be too disheartened by the team ending up in fifth, with Europa League football to look forward to.
Everton, meanwhile, finished seventh - one place above local rivals Liverpool - thanks to their seemingly now customary strong finish to the season.
Yesterday's win, which was marred after the final whistle by Cahill's sending-off following an altercation with Yohan Cabaye, saw the Toffees' unbeaten run in the league extend to nine games and Moyes took great satisfaction from the way his side had rounded off their campaign.
"I thought we looked like a side that could easily be pushing for a top four place," Moyes said.
"I don't think anyone could doubt that we were the side. Maybe we could have scored more. I take a good feeling that we've played well.
"We will try to build on it, start the season better and take it somewhere next year.
"I would hope people can see we're trying to progress, trying to play better, trying to get players in."
Newcastle boss Alan Pardew was disappointed to miss out on the Champions League spots but overall, rates 2011-12 as "one of the finest seasons at this club".
The Magpies, only promoted back to the top flight two years ago, have exceeded all expectation this term and Pardew - named as the Premier League's manager of the season last week - is determined to keep things on an upward trend going forward, although he admits it will not be easy.
"We have a more difficult season next season," Pardew said.
"We've got the African Nations Cup again, we also have the Olympics and the Europa League which will affect all our players.
"We will sit down this week to work out a gameplan to cover all that because it's not going to be easy.
"Next year, the focus is for me to try to improve us defensively as well as trying to get those extra few goals which give you 21 wins and a Champions League spot."