"With Peter on the story," Rather told The Early Show co-anchor Harry Smith, "you always knew you weren't going to sleep very much because you had to have your eye on him all the time. But you also knew how ethical he was and what a passion he had for news."
The Canadian-born Jennings, who announced in April that he had lung cancer,at his New York home. He was 67.
"Great pro, a loving husband and father, a loyal friend," is how Rather remembered Jennings, pointing out that the face of ABC News was one of the "most talented, caring and successful journalists, not just TV journalists, of all time."
It was Jennings' curiosity that partly made him that way, Rather said: "He was an intellectual in the best sense of the word. He had an unending curiosity about things."
Rather called it ironic that he was on assignment in Beirut when Jennings died. He said Jennings opened a bureau in Beirut for ABC news before anyone else had one there and that Jennings lived and reported from there in the 1970s and early '80s.
"It was part of his mission to keep the flag of international coverage (what we used to call foreign coverage) flying, and flying high. He felt it was so important for Americans and in his home country of Canada - he became an American - to understand what's going on in foreign countries.
"Peter played an exceptionally important role in keeping high quality international coverage at the top of the agenda."
Also important for Jennings was his family, Rather noted. The public rarely saw the side of the man who deeply cared for his wife Kayce Freed and his two children, Christopher and Elizabeth.
"He cared about his wife Kayce, one of the great journalistic couples," Rather said. "But I remember talking to Peter about his son Christopher once. He was so eager for Christopher to know about baseball. And Peter at that time - it had been some years ago - didn't know much about baseball. He said he wanted to take Christopher to a Yankee or Mets game."
Rather said the public also may not know how much Jennings cared about others, especially people less fortunate than himself.
Rather explained, "He had the jazz night in the Hamptons every summer, the proceeds of which went to help needy children."
As for the things Jennings enjoyed, Rather said, "He liked clogging. People would dance in those heavy shoes. I didn't know what it was. I only know the Texas beer joint two-step. And Peter was really into clogging. He liked the music of Harry Connick. There are so many good things to say about him. He was a lion of a man, a very decent man."