At a news conference Thursday, McGinn said he called Murray and congratulated him on his victory and pledged to work with him on the transition.
The latest election results show Murray, a state senator, with a significant lead over the incumbent.
The two candidates had largely campaigned with similar policy positions, but they offered contrasting styles of how to lead the Northwest's largest city.
Murray, who led the successful efforts to legalize gay marriage in the state, called for a more collaborative approach. McGinn was known for a combative style that rankled many in a city known for relatively tame political debate.
McGinn acknowledged Thursday that his style was off-putting to some, "but I hope people know I was always trying to do the right thing."
The first two years of his administration were marked by McGinn's vocal opposition to a tunnel replacement for the Alaskan Way Viaduct, a state highway bridge that runs through downtown Seattle. In a 2011 referendum that McGinn pushed for, voters endorsed the project, dealing him a significant political blow.
Later, he was involved in contentious negotiations with the Justice Department over changes at the Seattle Police Department, which DOJ investigators accused of routinely using excessive force.
McGinn, a basketball fan, used a sports metaphor to describe tactical missteps.
"I probably dribbled the ball off my foot a couple of times when I could've made a good pass instead," he said.
Before becoming mayor, McGinn was an activist with the environmental group the Sierra Club and continued to stake out a message of environmental stewardship.
He said Thursday that he was gratified that many of his positions on expanding transit options, broadband access and opposition to coal trains are now part of the mainstream in city politics.
Murray, who will take office in January, introduced his transition team leaders at an event Thursday afternoon. He chose Martha Choe, chief administrative officer for the Bill & Melinda Gates Foundation, and Dwight Dively, former city budget director.
"I think they reflect the type of leadership and the quality of folks we want to have as part of our transition team and part of our administration," Murray said of his team.
His campaign also released a statement saying Murray and McGinn had a "substantive and cordial" discussion Thursday morning.
"Ed thanked the mayor for his service, and told him that he respected his tremendous passion and dedication to the people of Seattle," the statement said.