The 23-year old striker has signed a four-year deal for an undisclosed fee which is believed to be around £700,000.
Russell, who scored 20 goals for the Arabs last season, had attracted interest from Scottish champions Celtic prior to joining the Rams.
"We're delighted to sign Johnny and his goalscoring exploits north of the border are something we've been aware of for some time," Rams boss Nigel Clough said.
"He's been prolific in the last few years and from what we've seen of him we believe he can transfer that form to the Championship.
"At 23 he is a player who's only going to improve as well and we feel he's the right type of player to complement the options and players we already have in attack.
"We've had some stiff competition from a number of clubs to get him here so it's nice to come out on top and convince him Derby County is the best place for him.
"Johnny is an ambitious and hungry player and he shares our desire to be fighting it out for a top six spot next season and we're pleased to have him on board."