Saying "the Electoral College has spoken," McConnell, speaking on the Senate floor, said, "this morning, our country has officially a president-elect and a vice president-elect."
"I want to congratulate President-elect Joe Biden," McConnell said.
After going through a litany of what he said were President Donald Trump's "many accomplishments," he turned to Monday's Electoral College vote.
"Many of us hoped that the presidential election would yield a different result, but our system of government has a process to determine who will be sworn in on January 20," he said.
"The president-elect is no stranger to the Senate. He's devoted himself to public service for many years," McConnell continued, before turning to Kamala Harris' historic win.
"I also want to congratulate the vice president-elect, our colleague from California, Senator Harris. Beyond our differences, all Americans can take pride that our nation has a female vice president-elect for the first time," he said.
Shortly afterward, Senate Minority Leader Chuck Schumer took the floor, saying, "For the sake of the country President Trump should take his cue from leader McConnell that it's time to end his term with a modicum of grace and dignity," adding, "enough is enough."
He said "Republican senators and Republicans throughout the country" should follow McConnell's lead and acknowledge Biden, Schumer said.
"Our Republican colleagues for the sake of our democracy, for the sake of the peaceful transition of power, should stop the shenanigans, stop the misrepresentations and acknowledge that Joe Biden will be the next president," Schumer said.
Speaking on the tarmac in Delaware ahead of his campaign trip to Georgia, Biden confirmed that has spoken with McConnell, saying they had a “good conversation” and expressing hope that they could work together in the future, saying he was “looking forward to it."
“I had a good conversation with Mitch McConnell today. We talked -- so I called to thank him for the congratulations, told him although we disagree on a lot of things, there’s things we can work together on. We’ve always been straight with one another, and we agreed we’d get together sooner than later. And I’m looking forward to working with him,” Biden said.
ABC News' Molly Nagle contributed to this report.