"It is essential that we provide the president with the tools and resources he needs to effect change, and that starts with putting a national security team in place as soon as possible," said Sen. John Kerry, D-Mass., chairman of the Foreign Relations Committee.
Obama's presidential rival, Sen. John McCain, was among those who spoke in Clinton's favor.
"This nation has come together in a way that it has not for some time," said the Arizona Republican, on the Senate floor for the first time since the inauguration.
Voters "want us to work together and get to work," McCain said.
As the Senate debated Clinton's appointment, Obama wasted no time in his first day at the White House. According to a White House spokesman, Obama placed telephone calls to Egyptian President Hosni Mubarak, Israeli Prime Minister Ehud Olmert, Jordan's King Abdullah II and Palestinian President Mahmoud Abbas.
The administration also planned to name former Senate Democratic leader George J. Mitchell as Clinton's special envoy for the Middle East. Dennis Ross, a longtime U.S. negotiator, was also expected to advise Clinton on Mideast policy, according to officials who spoke on condition of anonymity because they were not authorized to speak publicly about the move.
She was sworn in as the nation's 67th secretary of state in her office in the Russell Senate Office Building. Attending the private ceremony was her husband, former President Bill Clinton, and her Senate staff. According to her office, she used the Bible that belonged to her late father. To assume the position, she submitted her resignation as senator in twin letters to Vice President Joe Biden, as president of the Senate, and New York Gov. David Paterson.
The former first lady planned to report to the State Department on Thursday, where she was expected to address employees in the main lobby that morning - a tradition of sorts for secretaries of state on their first day on the job.
Clinton received overwhelming bipartisan support in Congress despite lingering concerns by some Republicans that her husband's charitable fundraising overseas could pose conflicts of interest.
Sen. Richard Lugar, the ranking Republican on the Foreign Relations Committee, proposed that former President Bill Clinton's foundation reject foreign contributions. But Hillary Clinton rejected Lugar's proposal, contending that the foundation's plan to disclose annually its list of donors and a range of its contributions already exceeds legal requirements.
Lugar said he hoped Clinton would re-examine her position but supported her appointment, citing her "remarkable qualifications" and "pressing global issues."
Vitter and DeMint, both members of the Senate Foreign Relations Committee, said they were unsatisfied. Last week, Vitter cast the sole opposing vote in the committee's 16-1 endorsement of Clinton. DeMint voted in favor of Clinton on the committee because he said he didn't want to obstruct a full Senate vote on her appointment, but ultimately did not support her nomination.
In addition to concerns surrounding the foundation, DeMint said he opposes Clinton's positions on such matters as providing aid to foreign groups that offer abortions.
"I do not plan to slow up this nomination, but I do find it difficult to support a nominee who I know will pursue policies so contrary to American sovereignty and the dignity of the human person," he said.
Following the vote, the Senate Foreign Relations Committee endorsed Susan Rice to become U.S. ambassador to the United Nations, a post Obama has elevated to the Cabinet level.
Meanwhile, the Senate considered other appointments by Obama. Treasury Secretary-designate Timothy Geithner apologized to the Senate Finance Committee and said he was careless for failing to pay $34,000 in Social Security and Medicare taxes earlier this decade. The committee planned to vote on his appointment on Thursday.
Eric Holder's bid to become the first African-American attorney general was delayed for at least a week when Republicans demanded more time to question him about harsh interrogations, Guantanamo trials and other topics.