The former England boss joins Derby from Queens Park Rangers, where he had been coaching since July, and Rush believes that he will give them the best chance of achieving promotion from the Championship.
"We are thrilled to announce Steve McClaren as our new head coach as well as Paul Simpson and Eric Steele in their respective roles. When we announced the departure of Nigel Clough it was stated that we were conducting a comprehensive review of the football structure, and we believe this is a major step forward in that regard," Rush told PA.
"Our goal is to have Derby County competing for a play-off place and beyond, and we feel with Steve and his team at the helm we have a great opportunity to do so. Steve is a highly qualified coach with all the skill-sets required to be a success here at Pride Park.
"He also has a network of contacts both on a European and global scale, and has also proved in the past that he can identify talent and develop it on the training ground, especially younger players, which is where our Academy comes into play and his working relationship with Darren Wassall is critical to our on-going success."
McClaren has past experience working with the Rams, having enjoyed two spells with the club first as a player and then as an assistant boss to Jim Smith. Rush admits that that was a factor in the decision to go for the 52-year-old.
"It is also important that Steve, Paul and Eric know what it means to play for Derby County and they have been at the club during the good times," he added.
"Steve was a driving force behind the club's promotion under Jim Smith in the 1990s and you only have to look at his CV to discover what a hugely talented coach he is.
"We are very much looking forward to letting Steve get on with the job in hand and helping Derby County move in the right direction."
McClaren will watch Tuesday's match against Ipswich Town from the stands before taking charge of the Rams for the first time on Saturday when they face Leeds United.