"Sounds like you want a steak, not an anchor", Roker commented, quoting the tweet in question.
Kotb will continue to co-host the show's fourth hour with Kathie Lee Gifford.
That makes them the first female team to lead the show ever - and just the second all-female team to ever headline a morning TV show since Diane Sawyer and Robin Roberts did it for ABC News in 2006 to 2009, "TODAY" wrote.
"It was such a shock to wake up one day and not have Matt, but it was the most natural and comforting thing in the world to have Hoda right there", Guthrie tells People, which features her and Kotb on its new cover.
Kotb had stepped in as guest host on the show in Lauer's absence. "Let's give her a round of applause", Guthrie said, grinning.
In an email announcing Kotb's new role, NBC News Chairman Andrew Lack stated that Kotb has "seamlessly stepped" into the co-anchor position, which she filled temporarily after Matt Lauer was sacked in November amid accusations of inappropriate sexual behavior.
For the first time in its decades-long history, "TODAY" will be led by two female co-anchors. "I think we should send some medics to Alexandria, Va, where my mom has likely fainted".
Ratings could explain the decision to keep Kotb as a host of the show.
The addition marks the first time a pair of women will be anchoring the program, according to NBC News.
"This is a very tough morning for both of us", Kotb said to Guthrie on the morning it was revealed Lauer had been fired.
"They have an undeniable connection with each other and most importantly, with viewers, a hallmark of TODAY", the email said, adding that Kotb was "remarkable".
According to USA Today, "TODAY" got a ratings boost the month after Lauer was sacked.
Ben Bogardus, a journalism professor at Connecticut's Quinnipiac College, said Kotb's selection was a good one "because it reinforces the sense of a family coming together after a shared trauma".