The contract is worth $14.55 million in a fully guaranteed deal with offset language, a league source told ESPN NFL Insider Adam Schefter.
Cooper gets $3.4 million within 10 days, $3 million by Sept. 15, and $2.56 million by Oct. 15 as the seventh overall pick.
The 6-foot-2, 311-pound guard from North Carolina missed the first two days of training camp. He had worked out with the starting unit through the offseason.
Arizona coach Bruce Arians got word of the contract agreement when a note was passed to him during his daily briefing with reporters on Sunday.
"Just got word that Jonathan Cooper is done and is on his way," Arians said. "He still missed time, but I'm a lot happier now. We're excited to have him in and watch him run the conditioning test -- in front of the whole team."
Although Cooper did miss a bit of training camp, his offseason work leaves him familiar with what's expected of him.
"He got a zillion reps," Arians said.
He joins an offensive unit that was called out for poor play by its coach on Sunday.
"That's the disappointing thing to me right now," Arians said. "It's not like our guys didn't have plenty of reps, because we had two practices going on during all of OTAs. There's no excuses right now for making the mental errors that we're making on offense."
The coach has declared competition for the two starting tackle jobs -- Levi Brown and Nate Potter on the left side, newcomer Eric Winston and Bobby Massie on the right. Daryn Colledge has moved from left to right guard with Lyle Sendlein at center.
"I'm very disappointed right now with the play up front," he said. "Way too many mental errors. The physical part we'll start seeing this afternoon. We've got to get better. I think that they've got enough pride that they're going to fix it. I think they're anxious to get pads on. Of all the people that get hurt playing in shorts, it's the big guys."
Cooper, a first-team Associated Press All-American, started a school-record 48 games at left guard for North Carolina and earned the 2012 Jacobs Blocking Trophy as the best lineman in the SEC. He is the first guard taken in the top eight of the draft since 1982, when Mike Munchak was selected by the Houston Oilers as the eighth overall pick.
In a draft loaded with players up front, Cooper was the fifth lineman selected.
In the end, the Cardinals did not have to set any precedent after the two sides wrangled over some contract language technicalities, general manager Steve Keim said.
And, Keim said, the team already is impressed with Cooper, as a player and a person, simply by the way he worked through the summer.
"Indications that you get from rookies most of the time can have variances, from good to bad," Keim said. "Early indications on Jonathan Cooper through OTAs and minicamp so far have been outstanding, almost to the point I would say better than advertised."
Information from The Associated Press was used in this report.