Senate Minority Leader Mitch McConnell (R-Ky.), who famously said that his main job was to ensure that President Obama was a one-term president, offered "sincere congratulations" to the president, yet suggested a stiffening resolve to challenge him.

Senate Minority Leader Mitch McConnell (J. Scott Applewhite -- Associated Press, File)

“The American people did two things: they gave President Obama a second chance to fix the problems that even he admits he failed to solve during his first four years in office, and they preserved Republican control of the House of Representatives," McConnell said in a statement. "The voters have not endorsed the failures or excesses of the President’s first term, they have simply given him more time to finish the job they asked him to do together with a Congress that restored balance to Washington after two years of one-party control."

Another Republican senator had a different take on the election. Sen. John Cornyn of Texas said that the Republican party has some soul-searching ahead. 

"But it’s clear that with our losses in the Presidential race, and a number of key Senate races, we have a period of reflection and recalibration ahead for the Republican Party," Cornyn said in a statement, released by National Republican Senatorial Committee. "While some will want to blame one wing of the party over the other, the reality is candidates from all corners of our GOP lost tonight.  Clearly we have work to do in the weeks and months ahead." 

McConnell ultimately said that Obama must move to the center, where Republicans would "be there to meet him halfway."

"That begins by proposing a way for both parties to work together in avoiding the ‘fiscal cliff’ without harming a weak and fragile economy, and when that is behind us work with us to reform the tax code and our broken entitlement system," he said.

Rejuvenated Obama wins second term

Romney's belief in self never wavered

Republicans awaken to grim future

50 States: Results and analysis