The House voted 399 to 17 yesterday to approve legislation making the Veterans Administration, the largest independent agency of the federal government, a department with Cabinet-level status.

The measure, which received a boost last week with the endorsement of President Reagan, goes to the Senate, where it has bipartisan support from conservatives and liberals alike. Senate committee hearings originally set for February have been moved up to next month.

The change has been actively sought by advocates of the nation's 27 million veterans, including the major veterans' organizations, who contend it would increase the visibility and voice of veterans as well as the VA's benefit programs.

"It's a really great day for veterans," said Rep. G.V. (Sonny) Montgomery (D-Miss.), chairman of the House Veterans' Affairs Committee. "It's a long time in coming . . . . But we're on a fast track now and that's great."

Rep. Clarence E. Miller (R-Ohio) told the House the bill was a fitting congressional tribute "to those who have suffered all, dared all and given all."

Rep. Jack Brooks (D-Tex.), the bill's floor manager, said the measure has been introduced in every Congress since he was first elected to the House 35 years ago.

Reagan's surprise support for the idea was announced to a veterans' group last week and came in spite of his frequently voiced concerns about the growth of government -- and his previous opposition to the two Cabinet-level creations of his predecessor, the departments of education and energy.

Rep. Patrick Williams (D-Mont.) called it "somewhat bizarre" that the VA department bill had such support when only a few years ago Congress wrangled at length over the Education Department bill and passed it only narrowly.

If enacted, the measure would turn the VA into the Veterans Affairs Department, and the VA administrator would become secretary and a member of the president's Cabinet.

It would make the top dozen jobs in the new department presidential appointments subject to Senate confirmation. VA Administrator Thomas K. Turnage would get a $10,000 raise if he were to become department secretary.

Brooks said Cabinet-level status for the VA would enable Congress to improve its monitoring of enforcement of laws affecting veterans. He said this direct oversight might have averted recent foreclosure of home mortgages held by veterans in southeastern Texas, if the VA had been prodded to intervene and save the veterans' housing under existing authority.

Supporters of the bill said it wouldn't involve any significant increase in federal spending or the size of the bureaucracy. The VA has estimated the cost at $30,000, covering pay raises for top VA officials. The Office of Management and Budget said it had no estimate of the cost.

Rep. Gerald Solomon (R-N.Y.) said Sen. John Glenn (D-Ohio) has agreed to hold Senate hearings next month, advancing them from February as scheduled.

With a $27 billion-a-year annual budget, the VA employs more than 240,000 people, making it second only to the Defense Department. It operates the largest health care system with 172 hospitals and hundreds more clinics and nursing homes. It also runs 111 national cemeteries and administers VA pension, compensation, home-loan guarantee and life insurance programs.