The vote to advance the measure was 73-25, 13 more than the 60 needed.
The outcome was not a surprise, and even some of the bill's opponents said they were satisfied they had been given ample opportunity over past weeks to try and give the bill a more conservative cast.
CBS News correspondent Bob Fuss reports that after more amendments, the bill will likely pass Thursday. But then comes the hard part – reconciling it with an enforcement-only bill that passed the House, declaring everyone here without papers a felon and making it a crime to help them.
Meanwhile, about 800 National Guard soldiers will head to the U.S. border with Mexico next week.
They will be the first of roughly 6,000 troops who will help the Border Patrol stem the flow of illegal immigrants across the border. The head of the National Guard told lawmakers that 200 soldiers will go to each of the four border states: California, Texas, Arizona and New Mexico.
Lt. Gen. Steven Blum said the initial group will be part of a force of project managers who will stay longer than the planned 21-day missions to provide continuity in the new border program.
President Bush has repeatedly urged Congress to approve an immigration bill that generally follows the approach taken by the Senate.
Senate Majority Leader Bill Frist, R-Tenn., predicted before the vote that the bill would receive "not overwhelming support but very strong support" and that a legislative compromise would be reached with the House.
"The problem is too big, with millions of people coming across the border and with hundreds dying as they come across the border," Frist said on NBC's "Today" show. "We as a governing body cannot simply turn and look the other way and say we're not going to do anything about it."
"It looks very much like the bill is on a path to conclusion," said Sen. Mel Martinez, R-Fla., one of the authors of the compromise bill.
On Tuesday, the Senate called for tougher employer penalties on businesses that hire illegal workers. The vote was 58-40.
Employers who do not use a new computerized system could be fined $200 to $600. The system would include information from the Social Security Administration, the Internal Revenue Service and Homeland Security Department.
There would be $20,000 fines for hiring illegal immigrants once the new screening system is in place, double the current maximum. Repeated violators could be sentenced to prison terms of up to three years.
Congress passed employer sanctions as part of the 1986 amnesty law, but they were never fully enforced and workers and employers got around them with fraudulent documents.
The Senate bill requires employers to check Social Security numbers and the immigration status of all new hires within 18 months after money is provided to the Homeland Security Department to expand the electronic system for screening workers.
Workers' information would have to be submitted to the electronic system within three days after the worker is hired. The Homeland Security Department would have to confirm the worker is legal or tell the employer the worker can't be immediately confirmed as a legal worker within 10 days.
The measure provides workers opportunities to contest the system's determination and to correct information that may be incorrectly flagging them as illegal workers. It also protects employers from liability if the screening system makes a mistake.
"This is probably the single most important thing we can do in terms of reducing the inflow of undocumented workers, making sure we can enforce in a systematic way rules governing who gets hired," Sen. Barack Obama, D-Ill., said Tuesday.
Opponents said the verification system would take years to implement and complained that workers deemed illegal could still hold onto jobs until their appeals are exhausted.
The House passed a bill in December that would impose fines on employers of undocumented workers ranging from $5,000 to $40,000. But, unlike the Senate bill, the House measure would require employers to screen all employees — an estimated 140 million people – instead of only new hires.
"I think working out a compromise will be very tough," William Frey of the Brookings Institute said. "I think at the very best there is a 50 percent chance of that happening."
In other immigration-related news, New York City Mayor Michael Bloomberg Wednesday called for a plan to establish a DNA or fingerprint database to track and verify all legal U.S. workers.
The mayor also said elements of the legislation moving through Congress are ridiculous, and said some lawmakers are living in a "fantasy."
In an editorial for The Wall Street Journal and an interview on CNN, the mayor said this country's 12 million illegal aliens should be given the opportunity for permanent status. He said that deporting them is impossible.
At an appearance in Staten Island later in the day, the mayor compared his proposed federal identification database to the Social Security card, insisting that such a system would not violate citizens' privacy.
The mayor, a former CEO of a financial information company, went on to say that if the technology is available, it should be used to create a worker ID database that will "uniquely identify the person" applying for a job, ensuring that cards aren't illegally transferred or forged.