The teams announced the trade Sunday and said the deal is pending physicals.
Quinn's departure comes one day after the Browns agreed to terms with free agent Jake Delhomme on a two-year contract. Quinn went 3-9 in 12 starts for Cleveland, which drafted him with the 22nd pick in the first round in 2007 out of Notre Dame.
"I appreciate everything Brady did for us last year and in his three seasons with the Cleveland Browns," coach Eric Mangini said in a statement. "He is professional in the way he goes about doing his job and worked extremely hard at every aspect of his game. I wish him the best of success in Denver."
Quinn won Cleveland's starting job last season, but was benched after 2 1/2 games for Derek Anderson. Quinn eventually got his job back when Anderson struggled, but he was hindered by inaccuracy.
He completed only 53 percent of his passes last season for 1,339 yards with eight touchdowns and seven interceptions. His rating was 67.3.
Quinn's days in Cleveland appeared numbered when new president Mike Holmgren failed to endorse him as the Browns' future starter. The Browns acquired backup Seneca Wallace in a trade with Seattle last week and have also agreed to a two-year deal with Jake Delhomme, a league source told NFL Network insider Jason La Canfora.
Instead, they dealt Cutler to Chicago for a bevy of draft picks and Orton, who beat out Chris Simms for the starting job in June.
That's the longest drought in the 26 years Pat Bowlen has owned the team. Bowlen recently said he likes Orton as his quarterback but would like to see the Broncos draft a passer next month.
Those plans may now be on hold.
The Broncos lost confidence in Simms last season, so it would appear he's out of the picture with the acquisition of Quinn.
Last week, the Broncos offered Orton, a restricted free agent, a first-round tender offer that would pay him $2.261 million this season.
Like hundreds of other players caught up in the league's labor issues, however, Orton is expected to boycott his team's start of offseason training, which starts Monday for the Broncos.
With the league ripping up the labor deal in hopes of getting a more owner-friendly contract, dozens of players who were set to reach the riches of unrestricted free agency this month are instead bound by their teams, which are offering much less than the players could have made in an unfettered marketplace.
Hillis will also be getting a fresh start.
McDaniels insisted that the problems were on the line and not in the backfield and that he'd rather give the ball to his top draft pick. With Spencer Larsen banged up for much of the season, McDaniels said his hands were often tied because he didn't want to risk running Hillis, who was often his only available fullback.
Hillis had just 13 carries for 53 yards last season. He was inactive for two games, once to attend his grandmother's funeral in November. He also returned four kicks.
The 240-pound bone-rattler energized the Broncos in 2008 when he emerged during an injury epidemic among the team's tailbacks and ran for a team-high 343 yards and scored six touchdowns before tearing his right hamstring against Kansas City in December.
The Browns seemed to have found their No. 1 tailback late last season in Jerome Harrison, who finished with a team-high 862 yards rushing. But veteran Jamal Lewis, second on the team in rushing, was released last month, which could provide an opening for Hillis to be the power complement to the small and quick Harrison.
So Hillis gets another chance in Cleveland while Quinn tries to jump-start his career in Denver.
The Associated Press contributed to this story.