The Browns forfeit their 2013 second-round pick in exchange for picking Gordon, the only player selected in the draft.
"We hope he comes in and plays right away," Browns general manager Tom Heckert told reporters in a conference call on Thursday, according to NFL.com. "That's the goal. Obviously, we'll see him here in a couple weeks, but a lot will have to depend on how he picks things up ... but, yeah, we expect him to play."
The 6-foot-3, 224-pound Gordon, who visited Cleveland last Thursday and Friday, ran a 4.52 40-yard dash in front of representatives from 21 teams at his pro day Tuesday, according to the Houston Chronicle.
"I enjoyed my visit there last week, and was very impressed with all of the people I met, including Mr. Heckert," Gordon said. "I have also had the chance to speak with Coach Shurmur on several occasions and I am looking forward to the opportunity to play for him."
Heckert had targeted wide receiver Kendall Wright, Gordon's former college teammate, in April's NFL draft. But after Wright was taken, the team selected quarterback Brandon Weeden, who will now have another target to throw to.
Gordon caught 42 passes for 714 yards and seven touchdowns in 2010.
"Anytime you pick somebody that early, you're hoping he's a good football player, and that's what we think Josh is," Heckert said. "... Most years, there's not a player of Josh's caliber in the supplemental draft, so you don't hear about it very much. But we look at it as a positive. We're getting a really good football player and we're getting him a year early."
Cleveland also has a rookie running back in Trent Richardson, the third overall pick in April's draft, to go with Weeden and Gordon.
Gordon is the first player by Cleveland taken in the supplemental draft since Bernie Kosar in 1985.
"Being taken this high in the supplemental draft and mentioned among the likes of the Bernie Kosar is a humbling experience in itself, as well as a dream come true," Gordon said.
Gordon spent last season out of football after Baylor dismissed him from the team in August 2011. Gordon was suspended at Baylor for a failed marijuana test, he told the Chronicle.
"It was against school policy, of course, and I was (suspended) in the summer," Gordon said. "I've definitely put that part of my life behind me. I don't plan to ever go back there. It was a difficult time, but I learned from it, and I've moved on."
Gordon transferred to Utah but had to leave the school for the supplemental draft for financial reasons.
Information from The Associated Press was used in this report.