Mr. McConnell’s view clashes violently with that of Senate Democrats and their allies around the country. They have viewed the beginning of the Trump era as a disaster best exemplified by an immigration executive order they decry as unconstitutional and un-American, as well as a selection of cabinet choices they rate as unqualified and carrying the very same baggage that has prevented others from being confirmed in the past.

Even some Republicans have criticized as inept and amateurish the rollout of the immigration order, which is now at the center of a federal court fight. Others have expressed trepidation at the prospect of being hammered in a Trump tweet if they run afoul of the new president.

Mr. McConnell, who is known for being able to take the temperature of his colleagues and to act accordingly, said he sensed no real unease about Mr. Trump on the Republican side of the aisle.

“We have had very good unity on our side,” Mr. McConnell said. “People are genuinely excited about taking the country in another direction. I don’t find any decision that he has made surprising.”

He said Senate Republicans had been enthusiastic about Mr. Trump’s cabinet nominees, dazzled by his Supreme Court pick and elated at the chance to roll back what he called the regulatory rampage of the Obama administration.

“A lot of us were wondering, what is Trump really going to be like?” Mr. McConnell said. “He used to support Democrats and have various views earlier in his life about politics. But when he got to the point of actually having the office and making the decisions, I think the decisions have been very comforting to my members, most of whom are a little bit right of center and further right of center.”

It goes without saying that having Mr. Trump in the White House gives House and Senate Republicans the opportunity to pursue an aggressive legislative agenda if they can find common ground. That prospect should be heightened by installing colleagues like Senator Jeff Sessions of Alabama and two fellow Republicans, Representatives Mick Mulvaney of South Carolina and Tom Price of Georgia, at the highest levels of the administration. Mr. Sessions is Mr. Trump’s nominee for attorney general while Mr. Mulvaney has been tapped to run the White House Office of Management and Budget and Mr. Price to head the Department of Health and Human Services.