The Bosnia international signed from Ferencvaros for around £4million last week, and made his first appearance in a blue shirt when Everton played Porto on Sunday.
Besic played 45 minutes of the game to mark Osman's years of service at Goodison Park as Steven Naismith's first-half strike was cancelled out by Jackson Martinez's effort just before the hour mark.
Speaking after his new signing's performance, Blues boss Martinez talked up the display from the midfielder, and stated that Besic looked as though he had been an Everton player for some time.
He told the Liverpool Echo: “I thought he looked like a player who has been at Everton for years. That’s what I saw in the World Cup – someone who is fazed by nothing, whether that’s man-marking Lionel Messi or getting on the ball and making you play. He’s a complete footballer.
“He’s only 21 and has the maturity of a very experienced player. I enjoyed his performance and even more so straight away the fans adopted him and made him feel like an Evertonian. That’s the best thing that can happen to a new signing. There was a real chemistry.
“I’m confident he’ll be fit for the start of the season. I’ve got no doubts he’ll cope with the Premier League. He’s exciting to work with and brings us something we haven’t got.”
It was a week to remember for the Blue half of Merseyside as Besic's arrival was followed up with the news of Ross Barkley's four-year contract extension.
Everton then announced the signing of Romelu Lukaku from Chelsea for a club-record outlay of £28 million on Wednesday, and while the powerful Belgium international will not be involved in the club's pre-season activities, Martinez doesn't feel it will hinder his marquee addition when the Premier League gets underway.
“He’s a young man,” Martinez added. “It’s when you’re over 30 that your body needs to go through specific routines. The period he’s had is long enough for him to refresh but not long enough for him to detrain. He’s not going to be involved in Germany as I make sure he’s ready for the start of the season.”
The Blues manager is reportedly still looking to add a couple more players to his ranks this coming season, and a loan deal for Chelsea winger Christian Atsu is said to be close.
Everton begin their Premier League with a trip to newly-promoted Leicester City on August 16.