"I am mindful that I follow the directorship of a distinguished CIA veteran, consummate intelligence officer, and devoted patriot," Pompeo said of Brennan in the message that was posted on the CIA's website.
Brennan's former deputy chief of staff, Nick Shapiro, said the ex-CIA chief and 25-year agency veteran had been "deeply saddened and angered
" by Trump after the commander-in-chief addressed hundreds of CIA employees at their headquarters in Langley, Virginia.
Trump spent much of his speech -- which he gave in front of a memorial wall that honors the 117 CIA officers who have fallen in the line of duty -- focusing on the size of the crowd at his inauguration, how many of those in the audience had voted for him, his appearances on magazine covers, and saying he "has a running war with the media."
"Former CIA Director Brennan is deeply saddened and angered at Donald Trump's despicable display of self-aggrandizement in front of CIA's Memorial Wall of Agency heroes," Shapiro said in a statement issued hours after Trump's visit to the headquarters. "Brennan says that Trump should be ashamed of himself."
Kellyanne Conway, a counselor to Trump, said Sunday on ABC that Brennan "sounded like a partisan political hack."
Due to his service on the House Intelligence Committee as a Kansas congressman, Pompeo crossed paths with Brennan many times during the latter's directorship of the agency.
And Pompeo's admiration of his predecessor is mutual.
Brennan told NPR last month: "I look forward to being able to hand this baton off to somebody who is as dedicated American as Mike Pompeo is, a former military officer and a serving member of Congress and somebody who is savoring the opportunity to come here to CIA."
In his note to the agency's employees, Pompeo called his being sworn-in as director "the honor of a lifetime," adding that he had spoken to "nearly every living former director" about the job.
"They all said it was the best job they had ever had," he said, later offering prayers for President George H.W. Bush, the 41st president and one-time CIA chief, and his wife Barbara, for a speedy recovery.
"President Bush told me that being Director was his second favorite job, but I did not really believe him," Pompeo wrote.