Ms. Sanders, who until Friday had served as a deputy press secretary, also managed Mr. Huckabee’s brief 2016 campaign before joining Mr. Trump’s White House bid. A mother of three young children whom she often mentions in tense moments in the White House briefing room, she inherited her father’s folksy style, his Christian conservatism and his biting sarcasm.

“If you want to see chaos,” she told a reporter on Friday, when asked about the West Wing turmoil, “you should come to my house early in the morning, when my three kids are running around. That’s chaos; this is nothing.”

In their back-to-back performances from the briefing room, Ms. Sanders and Mr. Scaramucci pointed up the contrasts of the Trump White House, where Washington veterans vie with New York insiders and family members for personal clout and control of the president’s message.

“It’s not clear yet what the overall effect will be. But what is clear is that to the extent that there’s a New York-versus-Washington dichotomy here, New York is winning,” said Anita Dunn, a onetime communications director for former President Barack Obama. “For me, the question is, is this going to help the White House move its agenda forward, and the jury is very much still out on that.”

Ms. Sanders began Friday’s session with a recitation of talking points and statements from the president before ceding the podium to Mr. Scaramucci, who launched into a 33-minute question-and-answer session with reporters, repeatedly assuring them he would get to every question as he bantered confidently with a press corps that has lately become more accustomed to lectures.

“I’m going to get to everybody,” he said repeatedly.

Mr. Trump announced Mr. Spicer’s departure and Mr. Scaramucci’s arrival in statements that Ms. Sanders read verbatim from the podium. But in a break with custom, it was left to Mr. Scaramucci, not the president, to announce that Ms. Sanders would be his new press secretary. The slight did not go unnoticed by veteran communicators, who said it could undercut her at the outset.

“It would have been appropriate for him to announce Sarah,” said Ari Fleischer, who served as press secretary for President George W. Bush. “It empowers her, it sends the message that the president appointed her, he’s standing next to her, and that signal was not there.”