An effort by moderate Republicans to rescue nearly $1 billion from President Reagan's budget cleaver failed yesterday when enough Democrats voted with the president to give him one of his clearest victories yet in the congressional budget battle.

By a vote of 59 to 40, with 16 Democrats joining most Republicans in upholding Reagan's position, the Senate rejected a proposal by Sen. John Chafee (R-R.I.) to restore $973 million that Reagan proposed to cut from a variety of social programs.

Chafee picked up 10 other Republican votes, despite administration lobbying against his effort and opposition from Budget Committee Chairman Pete V. Domenici (R-N.Mex.) as well as Majority leader Howard H. Baker Jr. (R-Tenn.). The administration's interest in defeating Chafee's proposal was underscored when Vice President Bush, who rarely presides over the Senate, took his chair for the vote.

But Chafee's cause became hopeless when the 16 Democrats, mostly from the conservative wing of the party and from sun-belt states that would benefit least from Chafee's proposals, refused to go along with their party leaders in supporting Chafee's cause.

During the past four days, Democrats -- at times even including some conservatives -- made repeated efforts to add money, often only modest amounts, for social programs that Reagan had marked for elimination or reduction in his budget-cutting drive.

Because Republicans had maintained a high degree of unity in opposing the Democratic initiatives, the breakaway effort by the Republican moderates was viewed as the best chance for restoring money for social programs in the first phase of Senate action on Reagan's budget proposals. As such, it had the support of Minority Leader Robert C. Byrd (D-W.Va.) and other Democratic leaders.

Most of Chafee's Republican supporters came from the Northeast and Midwest, giving the effort a regional cast, although the Chafee forces did their best to deny it, arging that they simply were trying to soften the blow of the budget cuts among people who could least afford the cost.

"We have met the test [of cutting the budget] and then some," argued Sen. John C. Danforth (R-Mo.) in noting that the Budget Committee -- and thus far the Senate -- had cut deeper into the budget than even Reagan proposed.

"And now," said Danforth, "we must mitigate the impact on those individuals and regions most harmed by what we've done."

Added Sen. Lowell C. Weicker (D-Conn.), "What's at stake here is not a political party, but human lives."

But Sen. Charles E. Grassley (R-Iowa) characterized the Chafee effort as, "too much business-as-usual, too much logrolling [and] not what the people expect of us."

Domenici, whom the Chafee forces had been trying to woo over to their cause, was less critical in opposing Chafee, saying he did so "with regret." He argued that there was no guarantee that the programs targeted by Chafee for increases actually would get them, because the resolution before the Senate simply instructs legislative committees to make a certain dollar amount of savings without specifying which programs should be cut in the process.

Chafee had proposed additional spending -- beyond what Reagan and the Budget Committee proposed -- of $450 million for education, $300 million for fuel assistance for the poor, $50 million for weatherization programs, $98 million for urban development grants, $50 million community and mental health and $25 million for mass transit.

"In my mind, it's a minimum of what the Senate should do. . . . It does not bust the budget," said Chafee.

Chafee's chances appeared somewhat improved earlier in the day when the Senate defeated a proposal by Sen. Bill Bradley (D-N.J.) to restore $3 billion for filling the Strategic Petroleum Reserve that the Budget Committee earlier this month had voted to cut. With that cut, the committee exceeded Reagan's proposed savings of $36.4 billion by $2.3 billion, Bradley's proposal was defeated, 81 to 17. Another proposal by Sen. James McClure (R-Idaho) to add $3 billion for the reserve and balance it off with a $3 billion cut from other programs failed, 76 to 22.

Also defeated by margins of roughly 2 to 1 were proposals by Sen. Daniel P. Moynihan (D-N.Y.) to restore $535 million for elementary and secondary education and for college loans for needy students. Also rejected was a proposal to continue funding of economic development assistance.

On Chafee's proposal, Maryland's senators voted yes, while Virginia's senators voted no