Romford-born Bunjaku, a Kosovo international, has agreed a two-year deal with the Rams having been on trial with the Sky Bet Championship club.
The 20-year-old will join his new team-mates for pre-season training when they report back at the end of this month and he will be eligible to feature for Lee Glover's Under-21 side during the forthcoming 2014/15 campaign.
Derby head of football operations Chris Evans told the club's official website: "Following a successful trial with the club, we are delighted to welcome Alban on board.
"He fits the criteria for the type of player we want to bring in for the development group.
"Alban has had a good grounding across Europe, having developed at both Arsenal and Sevilla.
"He will benefit immensely from working on the training ground in the coming months with Steve McClaren, with a view to nurturing his early potential."
Bunjaku joined Arsenal as an eight-year-old and progressed through the ranks at the Premier League club, scoring 15 goals in 23 starts for the Under-18 team before joining Spanish side Sevilla.
However, at the end of the 2011/12 season, Bunjaku was allowed to leave the Gunners for Sevilla on a free transfer, agreeing a three-year deal following a successful trial period with the La Liga outfit.