The clubs have also swapped later picks in this year's NAB AFL Draft as part of the deal, with the Demons giving up selection 45, but getting the Magpies' pick 58 in return.
The AFL will ratify the deal officially on Wednesday, with Dawes expected to receive $500,000 a season in a four-year deal.
Dawes announced his intentions on Friday to join the Demons, despite being contracted to the Magpies, where he played in the club's drought-breaking 2010 premiership.
Melbourne list manager Tim Harrington said the addition of Dawes was an important step in the club's rebuilding process.
"We see Chris as a long-term player for the Melbourne Football Club," Harrington said.
"He is a premiership player with plenty of finals experience and brings a much needed physical presence to our forward line.
"Chris fills a specific long-term need for our club. At 24 years of age, he will grow with the rest of our group and will help build a potent forward line set-up for the Melbourne Football Club."
Dawes left the Magpies on good terms.
"Chris can be proud of his contribution to Collingwood. He played some good football for the club, where he has been a popular figure, and shared in the 2010 premiership," Collingwood director of football Geoff Walsh said in a club statement.
"In his time with us, Chris always gave what he had and for that we thank him. We wish him well."
Trade tracker
The deal continues busy Gillette AFL Trade Periods for both clubs.
Melbourne has secured Dawes, as well as father-son recruit Jack Viney with its second-round draft selection.
It also snared 17-year-old power forward Jesse Hogan through the GWS mini-draft, although Hogan will not be able to make his AFL debut until 2014.
Young midfielder Dom Barry also joined the Demons as part of the Hogan deal.
Trade machine
Meanwhile the Magpies will take an unprecedented position into November's draft at the Gold Coast, now holding three first-round picks (17, 18 and 20).
It lost midfielder Sharrod Wellingham to West Coast for selection 17, but gained Eagles' forward Quinten Lynch as an unrestricted free agent. It is believed this acquisition was part of the reason for Dawes' departure.
Follow Callum Twomey on Twitter: @AFL_CalTwomey.
