WASHINGTON — She is a woman and an immigrant, a fixture of the Republican establishment for two decades. She is a savvy and professional practitioner of the capital’s inside game.

And now she is going to work for President-elect Donald J. Trump.

Mr. Trump named Elaine L. Chao on Tuesday as his choice to be the next secretary of transportation, elevating someone whose background and experience are in many respects completely at odds with the brash and disruptive tenor of his anti-Washington campaign.

But her selection also signaled Mr. Trump’s understanding of the need to surround himself with people who can help him accomplish the most ambitious parts of his agenda, even if they come from the political establishment he has so often scorned.

His transportation secretary is likely to be one of the more essential players. Mr. Trump, a real estate magnate, has said that infrastructure redevelopment will be a priority of his first 100 days in office. And Ms. Chao has experience — politically and personally — in navigating the competing centers of power in the capital. She is married to Senator Mitch McConnell, Republican of Kentucky and the majority leader.