From NBC's Ken Strickland
If the contested elections of the Minnesota, Alaska, and Georgia Senate races aren't resolved before the new Congress convenes in January, the Senate has the power to seat someone to the position until the matter is resolved. It's been done several times in Senate history, most recently as 1997 with Senator Mary Landrieu.
Article 1, Section 5 of the US Constitution states, "Each House shall be the Judge of the Elections, Returns, and Qualifications of it's own Members..."
How does it actually happen? The Senate Historian's Office explains it this way:
"...a petition has been presented to the Senate or a resolution offered by a senator contesting the election of a candidate (in some cases a year or more after the election in question). The contest may relate to the actual conduct of the election (vote count, electoral irregularities, etc.) or electoral misconduct by candidate or supporters. Most, but not all, of these cases were referred to a committee for review."
There has only been one case in Senate history when the chamber actually reversed the final election results. That was in 1926 in a race between Daniel Steck and Smith Brookhart in Iowa.
"Brookhart was initially seated but was later unseated by the Senate and Steck seated in his place," the Senate Historian's Office writes. "This is the only occasion to date in which the Senate has actually reversed the results of an election, unseated a senator, and seated the challenger."
The responsibility of making such a judgment would likely fall the Senate Rules Committee, if warranted.