Deulofeu's signing from Barcelona on a permanent basis was confirmed on Thursday.
In an interview with the Everton website, Jagielka hailed the qualities of the 21-year-old - who impressed at Goodison Park during a loan spell in the 2013/14 season.
Jagielka said: "He's always going to be a menace and create things and I'm sure that's why the manager has brought him in. It's a big thing with Gerard. As far as winning matches - getting those goals, getting those crosses in, beating people, winning free-kicks and things like that - he's something special.
"He did really well for us in his spell on loan and looked like he was going to kick on and get a chance at Barcelona. Obviously that didn't really happen but he's a fantastic player, he's got so much ability and he's a really good character, a really funny lad to have around the changing room.
"He didn't speak the most English when he came but tried his best and it's nice for the manager to be able to go out there and attract these players to want to come back."
Good deal for the Toffees?
The Spaniard managed to score four goals during his first spell at the club and everyone saw the talent he possesses, and it's testament to Roberto Martinez and the club that he wanted to head back to the club in order to get more game time, and try and further his career.
After a disappointing campaign in 2014/15, where they finished in the bottom half of the Premier League table, this is at least a signing for the fans to get excited about, and will give them a welcome pre-season lift.
Given he is older than when he first arrived at the club, and has more game time under his belt after spending last season with Europa League winners Sevilla, it will be interesting to see just how much he has developed, and whether he can have an ever greater impact on the side than he managed two years ago.