Following his appointment as Toffees manager over the summer, Martinez selected Jelavic in his first XI for the club's opening five Barclays Premier League games this term, but the frontman's involvement in the competition thereafter has amounted to only four appearances, all as a substitute.
He came into the team on Saturday for his first start since September and scored for the first time since March, netting twice as the Merseysiders defeated Sky Bet Championship promotion-chasers QPR 4-0 at home in the third round of the FA Cup.
And regarding Jelavic's performance, Barry, who is on loan with Everton from Manchester City for the season, said: "I think the manager deserves a lot of credit for that.
"It is important with a squad that you keep the players that haven't been playing hungry, and Niki is a great example.
"He has had to be really patient, waiting for his chance, and it is not easy to come in sometimes. But he has shown a great hunger and scored twice.
"Niki has been unfortunate with the way Romelu (Lukaku, the striker who is on loan at Everton from Chelsea and has nine goals for the Toffees this season) has come in and played, and with the way the team has been getting results.
"But the manager has kept everyone hungry and he has kept Niki smiling.
"At times it can be tough, but it showed (in the QPR game) that he still wanted to go and do a job for Everton."
Jelavic's career with Everton, who are currently fifth in the Premier League, began impressively, with him scoring 11 times for them in the second half of the 2011/12 season after joining in January, and he managed to net eight more Toffees goals last term.
Yet it seems his spell at Goodison Park could be coming to an end, with Martinez - mindful that the 28-year-old hopes to play at this summer's World Cup - having made it clear he thinks it is only fair he listens to offers for Jelavic.
There have been inquiries from other clubs, with QPR reportedly among them.
But Martinez has also said he does not really want to lose Jelavic's services, stressing he will not be sold "on the cheap", and while the player himself has indicated he is keeping an open mind about his future, he has also emphasised he is looking for more of a chance under the Spaniard.
Barry, meanwhile, sees it as a very happy camp overall at Everton, from which he would be surprised to see players depart during the transfer window.
The 32-year-old, who has 53 England caps, said: "We don't know what is going to happen with his (Jelavic's) future, but the fans have showed their appreciation for him, because he has done a great job for Everton - not just this season, but in years gone by.
"I'd be very surprised if we lose any players. For me, this is one of the best clubs to be learning your trade at at the moment - I think all the players will say it.
"They look forward to going to training and working with this manager, and the results have been showing that too."
In Martinez, Everton - without a trophy since the 1995 FA Cup - have a boss who won the competition last season as manager of Wigan.
The Toffees team that took on QPR featured Joel Robles, Antolin Alcaraz and James McCarthy, three players from that Latics triumph at Wembley in May, as well as Barry who won the FA Cup with City in 2011.
Certainly Everton have considerable relevant experience to draw upon as they look to break their own trophy drought in this year's FA Cup, and Barry said: "I think for any club, when you go so long without a trophy, it does weigh on you a bit and that first trophy is always the hardest one to get, as it showed with Man City a few years back.
"The FA Cup is a tricky competition - you need a lot of luck along the way. But if we produce performances like (they did against QPR), and with the squad we have, we have every chance of going all the way.
"We have not spoken about winning it yet to be honest. But there is enough experience there for us to go on a great run and you never know - the FA Cup can bring glory."
Everton face League One side Stevenage away in round four.
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