Chief Justice John Roberts administered the oath in a private ceremony at the Supreme Court building across from the Capitol at about 12:40 p.m., EDT, court officials said.
Alito was expected to join Roberts and the rest of the Supreme Court justices at tonight's.
Alito will be sworn in a second time at a White House ceremony Wednesday.
The final Senate vote was 58-42, with all but one of the Senate's majority Republicans voting for Alito's confirmation, and all but four of the Democrats voting against him.
CBS News correspondent Gloria Borger reported that sticking to party lines is going to stay the norm. In a meeting today with top congressional Democrats, Borger reports they expressed a desire to work with Republicans, but said it would be impossible in the current Washington climate, which House Democratic Leader Nancy Pelosi described as one of "corruption, incompetence and cronyism."
That is the smallest number of senators in the president's opposing party to support a Supreme Court justice in modern history. Chief Justice John Roberts got 22 Democratic votes last year. Justice Clarence Thomas, who was confirmed in 1991 on a 52-48 vote, got 11 Democratic votes.
Alito watched the confirmation vote on television at the White House. CBS News correspondent Mark Knoller reports that a cheer went up in the Roosevelt Room as the tally was announced confirming Alito.
President Bush led in the applause as he and Alito watched the roll call vote together. Mr. Bush shook Alito's hand and told him "now go to work."
In a written statement, Mr. Bush said he was "pleased" with the Senate vote, and he again described Alito as "a brilliant and fair-minded judge who strictly interprets the Constitution and laws and does not legislate from the bench."
Senate confirmation of Alito is a major political victory for the president in a midterm election year, CBS News correspondent Peter Maer reports. The political timing is good for Republicans because the addition of Alito to the high court will help the president appeal to core conservatives who turn out in off-year elections.
Alito is a longtime federal appeals judge, having been confirmed by the Senate by unanimous consent on the 3rd U.S. Circuit Court of Appeals in Philadelphia on April 27, 1990. Before that, he worked as New Jersey's U.S. attorney and as a lawyer in the Justice Department for the Reagan administration.
It was his Reagan-era work that caused the most controversy during his three-month candidacy for the high court.
Alito replaces Sandra Day O'Connor, the court's first female justice and a key moderate swing vote on issues such as assisted suicide, campaign finance law, the death penalty, affirmative action and abortion.
Barak Obama, a Democratic senator from Illinois, told CBS News that he had no doubt Alito is going to be more conservative than O'Connor.
"I hope that Judge Alito recognizes that he's got a role that's unique in America, where Supreme Court justices with lifetime appointments can change the landscape for this country and not subject to democratic checks," Obama said. "Hopefully he'll exercise his job with great and sober judgment."