"This is what any man would dream of leaving behind," Kotb said during a lengthy segment in the fourth hour of the show. Her partner, Kathie Lee, was notably absent as she mourned her husband.
Earlier on Monday morning, Kotb broke down as she talked about visiting Kathie Lee following Frank's death.
"She is remarkably strong, I'm in awe of her even today," Kotb, 51, said.
Later, she added: 'Kathie said something to me yesterday, she said 'This is not a tragedy. I was holding her hand and she said, 'A four-year-old girl having cancer is a tragedy.'
"She said Frank lived to be nearly 85, she said he lived an incredible life, has an incredible family, has an incredible legacy. She said his life is a triumph."
Gifford, a former New York Giants running back who became a legendary sportscaster, died Sunday of natural causes in his Connecticut home he shared with Kathie Lee, 61.
Kotb also thanked viewers for their "outpouring of love and affection" on Twitter and Facebook, and she announced that anyone who wants to help can donate to the nonprofit dedicated to disabled children that Kathie Lee is involved in.
"A lot of people want to know how they can help. Kathie Lee is asking for donations to Cassidy's Place in lieu of flowers," said Kotb. "They would love you to donate anything that you'd like to give them."
On Sunday, the family released a statement to PEOPLE announcing the football star's death.
The statement reads in part: "We rejoice in the extraordinary life he was privileged to live, and we feel grateful and blessed to have been loved by such an amazing human being."
Deeply grateful to all 4 ur outpouring of grace. We r steadfast in our faith & finding comfort in knowing where Frank is. Phillippians 4:13.— Kathie Lee Gifford (@KathieLGifford) August 9, 2015
Kathie Lee sent a grateful Tweet to fans on Sunday as well, thanking them for their outpouring of support. Kotb re-Tweeted a post from the Kathie Lee and Hoda Twitter account, which included a photo of herself and Kathie Lee with Frank.
"Today, we lost a member of the 4th Hour family," the Tweet reads. "Rest in peace, Frank Gifford. Our prayers are with @KathieLGifford."
very sad day. Rest in Peace to my Godfather. The legend, Frank Gifford.— Kendall Jenner (@KendallJenner) August 9, 2015
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Since Gifford's death, a slew of stars have commented on social media about the football icon's passing, including his goddaughter Kendall Jenner, Larry King and Al Roker.
Gifford is survived by Kathie Lee and their two children, Cody, 25, and Cassidy, 22, as well as three children from a previous marriage, Jeff, Kyle and Victoria.