The vote was 71-26, more than the two-thirds of those present and voting required for approval.
The treaty would cap nuclear warheads for both countries and resume onsite inspections that expired a year ago.
Ratification requires two-thirds of those voting in the Senate and Democrats need at least nine Republicans to overcome the opposition of Minority Leader Mitch McConnell of Kentucky and Jon Kyl of Arizona, the GOP's point man on the pact.
The treaty will leave the United States "with enough nuclear warheads to blow any attacker to kingdom come," Alexander said on the Senate floor, adding, "I'm convinced that Americans are safer and more secure with the New START treaty than without it."
"We are on the brink of writing the next chapter in the 40-year history of wrestling with the threat of nuclear weapons," Foreign Relations Committee Chairman John Kerry (D-Mass.) said after the vote.
But conservative foes of the accord - among them possible GOP presidential candidates Mitt Romney, Sarah Palin, Newt Gingrich and Tim Pawlenty - argue that the treaty would restrict U.S. options on a missile defense system and lacks sufficient procedures to verify Russia's adherence.
"The administration did not negotiate a good treaty. They went into the negotiations it seems to me with the attitude with the Russians just like the guy who goes into the car dealership and says, 'I'm not leaving here until I buy a car,' " Kyl said.
The treaty would limit each country's strategic nuclear warheads to 1,550, down from the current ceiling of 2,200. It also would establish a system for monitoring and verification. U.S. weapons inspections ended a year ago with the expiration of a 1991 treaty.