Franken, accompanied by his wife and amid cheers from supporters, said he had received a "very gracious call" from Coleman, who earlier conceded the Minnesota Senate race after the Minnesota Supreme Court ruled against him in an appeal over disputed absentee ballots.
Franken's final margin of victory over Coleman was just 312 votes out of 2.9 million cast.
Franken said he and his family are "so thrilled that we can finally celebrate this victory, and I am so excited to finally be able to get to work for the people of Minnesota."
"I know there's been a lot of talk about the fact that when I'm sworn in I'll be the sixtieth member of the Democratic caucus," Franken said, a reference to the fact that his presence in the Senate gives Democrats a filibuster proof majority. "But that's not how I see it."
"I'm going to Washington to be the second senator from the State of Minnesota, and that's how I'm going to do this job," he continued. He acknowledged, however, that he hopes to help push through President Obama's ambitious agenda.
Earlier, Coleman made an appearance at which he announced his concession and said "it is time now to move forward." The Republican could have appealed the Minnesota Supreme Court decision to federal courts but elected not to do so.
In his comments, Franken thanked Minnesotans for their patience and said he would fight for everyone in the state.
"I won by 312 votes, so I really have to earn the trust of the people who didn't vote for me," he said.
Franken said he had spoken to Senate Majority Leader Harry Reid, who said Franken will be on the Health, Education, Aging, Indian Affairs, Labor and Pension and Judiciary Committees.
Watch Senator-elect Franken Discuss His Victory: