In the op-ed, the staffer said senior officials considered trying to remove Trump under the 25th Amendment to the Constitution - which allows for a president's removal if a majority of the Cabinet and the Vice President deem him or her unfit to serve - but decided against doing so.
Four top White House officials, including Vice President Mike Pence, said on Thursday they did not write an anonymous column in the New York Times that described "a quiet resistance" to President Donald Trump within his own administration.
Trump called it a "gutless editorial", but his public anger was initially directed more toward The Times than the piece's author, whom he dismissed as someone who's probably "failing" at his job and is in the administration "for all the wrong reasons".
Sanders said the administration was "disappointed, but not surprised, that the paper chose to publish this pathetic, reckless and selfish op-ed".
The decision was in the purview of James Bennet, editorial page editor, and James Dao, op-ed editor, with publisher A.G. Sulzberger weighing in, a Times spokeswoman said.
"The dilemma - which he does not fully grasp - is that numerous senior officials in his own administration are working diligently from within to frustrate parts of his agenda and his worst inclinations", the official writes.
The author, saying "it may be cold comfort in this chaotic era", wanted Americans to know that "there are adults in the room".
"We fully recognise what is happening".
Woodward quotes unidentified White House aides who claimed to have saved the country by ignoring Trump's orders on several occasions and even stealing his documents. "We believe publishing this essay anonymously is the only way to deliver an important perspective to our readers", the editors at the paper wrote. "Can you imagine what it must be like to have a job at which nearly everyone who works for you thinks you're an idiot?" he added.
The author called Trump "amoral", as well as "generally anti-trade and anti-democratic". "What it really means is we don't have a president".
He continued on Wednesday: "The already discredited Woodward book, so many lies and phony sources, has me calling Jeff Sessions "mentally retarded" and "a dumb southerner".
As for the anonymous author, the press secretary accused the person of "not putting country first but putting himself and his ego ahead of the will of the American people".
In the op-ed piece, the unidentified senior administration official sounds an alarm about Trump's "amorality" and "impetuous" leadership style.
In the article entitled "I Am Part of the Resistance Inside the Trump Administration", the official stressed they were committed to the Republican agenda, and did not side with opposition Democrats.
Mr Trump unloaded against the Woodward book, labelling the tell-all memoir "a work of fiction" as West Wing staff scrambled to rebut its vivid depictions of White House dysfunction.
The writer remains anonymous, but it's not the first time we've heard that Trump Administration officials see themselves as protecting America from Trump.
President Donald Trump listens to Emir of Kuwait Sheikh Sabah Al Ahmad Al Sabah in the Oval Office of the White House, Sept. 5, 2018, in Washington.