Petrino has produced some of the most prolific offenses in the nation as an assistant coach at several other schools, Idaho officials said.
"He is one of the top offensive minds in college football," school president Duane Nellis said.
Petrino replaces interim coach Jason Gesser, who guided the Vandals during the final four games after Robb Akey was fired.
Terms of his contract were not disclosed.
"All I've ever wanted to do was to be a head coach," said Petrino, who is the brother of former Razorbacks coach Bobby Petrino.
Petrino said he would bring his high-powered offensive schemes to Idaho, and restore the program to the success it enjoyed two decades ago when he was an assistant here.
"People feared the Vandals," he said. "This program has a great tradition."
Petrino most recently served as the offensive coordinator and quarterbacks coach at Arkansas, where this year he went through the firing of his brother, who was dismissed for hiring his mistress to a position in the athletic department and initially lying about her presence during an April 1 motorcycle accident, and the more recent departure of interim Arkansas coach John L. Smith, a long-time mentor.
Paul Petrino served as an assistant coach at Idaho under Smith from 1992-94, and later coached at Utah State, Illinois, Louisville and with the Atlanta Falcons of the NFL.
He was a standout quarterback at Carroll College in Helena, Mont.
Petrino was one of five finalists for the job at Idaho, which is struggling to remain in the FBS after the Western Athletic Conference dissolved as a football league. The Vandals will play as an independent next year, and are still searching for a conference to join.
"We want to stay FBS," he said, and the best way to do that is to win games and make themselves attractive to a conference.
He was not ready to announce any of his staff.
Petrino declined to place a timeline on winning.
"We will look at getting better every day," he said. "Your goal has got to be a quick goal. I see guys getting fired after one year now."
Akey was fired in October with the Vandals 1-7 at that point. Gesser was appointed interim head coach and the Vandals lost their final four games. Akey was 20-50 in five-plus seasons as head coach.
Petrino said his father, Bob, his brother, Bobby, and Smith were the three main football influences in his life.
He noted that his brother taught him a lot about football.
"He still is the most brilliant coach in football," he said of Bobby. "He taught me so much about football, paying attention to every detail."