As he pressed his case last week, before the allied strikes with Britain and France, Mr. Mattis lost the battle over getting congressional authorization. But he won the larger war.

Mr. Mattis prevailed in limiting the strikes to three targets that did not risk endangering Russian troops scattered at military installations around Syria. Nor did the 105 missiles hit Syrian military units believed to be responsible for carrying out an April 7 suspected chemical weapons attack on Douma, near Damascus.

In the end, the narrowly targeted strikes belied Mr. Trump’s description Friday night of a larger coordinated response that could take days or weeks.

“The combined American, British and French response to these atrocities will integrate all instruments of our national power — military, economic and diplomatic,” Mr. Trump said in an address to the nation as the strikes were underway. “We are prepared to sustain this response until the Syrian regime stops its use of prohibited chemical agents.”

But there have been no additional strikes since then, and the Pentagon said no more are being planned. “This is a one-time shot,” Mr. Mattis said on Friday, calling the airstrikes “a very strong message to dissuade” President Bashar al-Assad of Syria from using chemical weapons against his own people.

Mr. Trump’s drumbeat of threats last week of a sharp response to the suspected gas attacks all but guaranteed that the United States military would strike Syria, according to two Defense Department officials who spoke on condition of anonymity.

Even so, Mr. Mattis pushed to get congressional authorization, according to people with knowledge of the internal debate. In several White House meetings last week, he underscored the importance of linking military operations to public support — a view Mr. Mattis has long held.