Marshall and the Dolphins subsequently agreed on a four-year, $47.5 million contract extension that includes $24 million in guaranteed money, according to a league source.
The deal includes close to $29 million in the first three years of the contract and makes Marshall the highest-paid receiver in NFL history. Arizona's Larry Fitzgerald previously had the richest deal of any receiver, averaging $10 million per season.
The trade is contingent upon the league processing the paperwork and Marshall passing his physical.
In acquiring Marshall, the Dolphins gained a deep threat at receiver to loosen up defenses for their potentially potent ground game and young, strong-armed quarterback Chad Henne. In return, the Broncos get two high draft picks for a player who caused them headaches in addition to making plays.
"It's good to be here, man," Marshall said as he mingled with fans at the Miami Heat's game against New Jersey.
Last summer, Marshall was suspended by the team for throwing a tantrum at practice during training camp. The source of his frustration was twofold: he was upset with the team's medical staff for misdiagnosing a hip injury that required offseason surgery and he was displeased with his contract.
Despite those difficulties, Marshall caught 101 passes for 1,120 yards and 10 touchdowns last season, including an NFL-record 21 in a loss at Indianapolis. He has 327 career receptions for 4,019 yards and 25 touchdowns in four seasons, all in Denver, which drafted him out of Central Florida with a fourth-round pick in 2006.
Broncos coach Josh McDaniels told ESPN's Ed Werder that in a post-trade phone call with Marshall, he told the receiver that he enjoyed their time together, predicted success for him with the Dolphins and expressed regret that their own partnership could not be successful in the long term.
Marshall's departure leaves the Broncos with Jabar Gaffney, who had 54 catches for 732 yards and two TDs last season, as their top receiver. Eddie Royal caught 37 passes with no TDs and Brandon Stokley had 19 catches, four for TDs.
"I'm not quite sure how the future would have been if he was around and how it would have affected everything," Broncos tight end Daniel Graham said at an event at a Denver high school on Wednesday night.
In a little more than a year, with a pair of blockbuster trades, Denver has recouped two first-round picks, two second-round picks, a third-round pick and quarterback Kyle Orton in exchange for Jay Cutler and Marshall.
"We're going to miss what he brings to that football field," Graham said. "But we've got to move on. It's a business."
Graham, who grew up in Denver, said he realizes a good portion of the team's fan base is upset that McDaniels has traded away Cutler and Marshall in successive offseasons.
"I know looking from the outside it's hard to understand what's going on, especially when you lose players the caliber we've lost in the past couple of years, but it is the ultimate team sport," Graham said. "Not one player is going to win the Super Bowl for us."
McDaniels told Werder the decisions to trade Cutler and Marshall were motivated by his belief that the moves were essential to build a successful team. He also said he is convinced that the Broncos achieved the best possible result in trading Marshall for a pair of second-round draft picks, while remaining mindful of Marshall's interests.
"It seemed like he really wanted to get out of here," Graham said. "He wanted to go somewhere where he felt he was going to be happy, and he didn't feel like that would be here in Denver."
The trade also shakes up things in the AFC East, as Miami has arguably topped the deal the New York Jets made earlier this week, when they traded a fifth-round draft pick for talented but troubled Steelers receiver Santonio Holmes.
Miami is a surprise destination for Marshall because Dolphins coach Tony Sparano hadn't shown much interest in acquiring him. But football czar Bill Parcells, who has final say on personnel matters in Miami, has never shied away from talented-but-troubled wide receivers, working with Terry Glenn in New England, Keyshawn Johnson in New York and Terrell Owens in Dallas.
The Dolphins' passing game can use the help.
Speedy Ted Ginn Jr. has been a disappointment since Miami took him with the ninth overall draft pick in 2007. Ginn and the other wideouts -- Brian Hartline, Davone Bess and Greg Camarillo -- combined for only six touchdown catches last season.
"We need big playmakers," Dolphins general manager Jeff Ireland said in February. "We need players that make chunk yardage. We need players that score touchdowns. We need to get more production out of the wide receiver position."
The acquisition of a top-line receiver frees up the Dolphins to use the No. 12 pick in the draft next week on the defensive front seven, where help is also needed.
The trade, a complex negotiation involving as many as four teams, began shortly after Marshall failed to secure a contract offer as a restricted free agent and signed his one-year tender with the Broncos on Tuesday morning. A source with direct involvement in the talks told Werder that McDaniels and general manager Brian Xanders negotiated with Parcells and Dolphins general manager Jeff Ireland into the early-morning hours Wednesday before reaching an agreement.
Miami was hot after Marshall, as was Seattle. Tampa Bay showed some interest, and even the Jets considered making a play for Marshall before making their deal for Holmes, according to sources.
Marshall, 26, is one of the game's best young wide receivers, a mercurial talent in the prime of his career at the position where the Dolphins needed help most. But like Holmes, he also brings a reputation of being difficult to handle and a record that has him one strike from a yearlong suspension.
Marshall has had several run-ins with the law and visits to commissioner Roger Goodell's office to discuss his off-the-field behavior. He was suspended for the 2008 opener following a series of domestic disputes involving a former girlfriend.
In February, an emotional Marshall was a witness in the murder trial of Willie Clark, who was convicted in the 2007 drive-by slaying of Broncos cornerback Darrent Williams.
Adam Schefter is ESPN's NFL Insider. Information from ESPN senior NFL analyst Chris Mortensen, ESPN NFL reporter Ed Werder and The Associated Press was used in this report.