Trump told journalists he "really" appreciated the pair's statements, which helped show that "the book is just a piece of fiction".
The Watergate legend said the president called him last month saying he wished he had been interviewed for the book.
While neither former staffer directly denied details in the book, Porter said in a statement that the book offers a "selective and often misleading portrait".
Several administration officials, including Defense Secretary James Mattis and chief of staff John Kelly, have denied the anecdotes and quotes ascribed to them in Woodward's book.
Trump has dismissed the Woodward book as a "scam" filled with "made up" quotes.
The president blasted Brzezinski as "low I.Q. Crazy Mika" and said she had been "bleeding badly from a face-lift" during a New Year's party at his Florida resort.
"Fear: Trump in the White House" hits United States bookstores on Tuesday after a weeklong buildup, with published excerpts, leaks and interviews that portray Trump as dangerously erratic and uninformed.
That claim aside, people inside the West Wing say the identity of the writer remains a topic of heated discussion as reports emerged that a list of possible authors had been drawn up.
Then-economic adviser Gary Cohn reportedly told Trump that he couldn't raise the top tax rate for the simple reason that he was Republican and would "get absolutely destroyed" if he did so.
"This has not been treated seriously enough", Woodward said on "Today".
"The secretary of defence has to tell him: "because we're trying to prevent World War Three".
Asked later whether the White House would list all the things in Woodward's book that are wrong, Sanders said such an exercise "would be a complete and utter waste of our time, so no".
The Washington Post, which obtained an advance copy of the book by the veteran chronicler of modern presidents, reported Tuesday that Woodward describes Trump manically pressing his staff for actions that could lead to major conflict - leaving them little choice but to disregard his orders.
"One key person who's in office called me and said, "Everyone knows what you said here is true".
"If that person had come to me while I was working on this book or if it were someone I interviewed I would say, 'I need specifics.' The building blocks of journalism, of truth, are specific incidents".