Rex Tillerson didn’t have the easiest ride to confirmation, but it will likely be smoother than his arrival at the State Department, where the nation’s new top diplomat confronts a complex slate of foreign policy challenges — in addition to a tricky management challenge overseeing a pool of career diplomats angry over President Trump’s executive order on immigration policy and refugees.

The Senate voted 56-43 to confirm Mr. Tillerson as the nation’s 69th secretary of state on Wednesday, with Republicans picking up three Democratic votes to pierce the minority’s hoped-for united front against the former ExxonMobil CEO.

The 43 negative votes are believed to be the most ever against a secretary of state nominee.

Mr. Tillerson, who lunched with the president just hours before the Senate vote, was sworn in Wednesday evening at a White House ceremony with President Trump and Vice President Mike Pence. Mr. Trump, in brief remarks, praised Mr. Tillerson’s background and experience, saying, “Your whole life has prepared you for this moment.”

Noting the outpouring of encouragement and advice he received from friends and strangers since his surprise selection, Mr. Tillerson vowed to “always represent the interests of all the American people at all times.”

The 64-year-old Texan will take the helm in Foggy Bottom at a moment of uncertainty and antipathy among many world leaders over Mr. Trump’s controversial executive order and his “America First” approach, which has added to the already turbulent landscape of foreign policy challenges — from the North Korean nuclear threat to Syria’s civil war, a clash with Mexico over trade and border issues, Russian meddling in Ukraine and the fight against the Islamic State.

Hundreds of Foreign Service officers and State Department rank and file have reportedly signed a “dissent memo” in recent days criticizing the order Mr. Trump signed Friday to temporarily suspend all refugee access to the U.S. and halt visas to citizens of seven Muslim-majority nations.

White House spokesman Sean Spicer on Monday advised the restive diplomats to “either get with the program or they can go,” but took a softer line Wednesday, suggesting an awareness at White House of Mr. Tillerson’s delicate task boosting morale and support as he takes his new job.

“Every American has a right to speak their mind,” Mr. Spicer said, adding that Mr. Tillerson “has a very open mind and a desire to meet with people and hear ideas.”

Mr. Tillerson was noncommittal on the visa and refugee issue during his nomination hearing last month. While he voiced apprehension toward Mr. Trump’s campaign calls for a ban on “all Muslims” entering the U.S., he said he might be open to the creation of some kind of registry of Muslims living in the country.

All of the Senate’s 52 Republicans voted to confirm Mr. Tillerson Wednesday, with Democratic Sens. Joe Manchin III of West Virginia, Heidi Heitkamp of North Dakota, Mark R. Warner of Virginia, and Democratic-leaning independent Sen. Angus S. King Jr. of Maine joining the GOP majority. Mr. Manchin and Ms. Heitkamp are up for re-election next year in states that President Trump easily carried in November.

House Foreign Affairs Committee Chairman Ed Royce said that “having a world-class manager at the State Department will be a real asset,” but it remains to be seen how Mr. Tillerson’s experience running a massive oil company will translate will translate at State.

One key will be Mr. Trump and Mr. Tillerson’s choice to be deputy secretary of state — a position essential to managing both the message and policy implementation across the department’s many bureaus and missions focused on different parts of the world.

Several names have been floated, including Paula Dobriansky, who served as an under secretary in the George W. Bush administration, and Elliott Abrams, who held various assistant secretary positions under Mr. Bush and President Reagan.

One State Department official, speaking on background on Wednesday, said many at the State Department believe Mr. Tillerson and Mr. Trump are keen to reorganize the department’s management structure and pare down the politically appointed positions that have swelled in number since the late 1970s.

Critics say Mr. Tillerson lacks any diplomatic experience and, like Mr. Trump, has a history of amiable relations with Russian President Vladimir Putin.

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