The Senate, bowing to the lure of Saturday mail delivery and veterans' benefits, last night approved a $613.1 billion budget plan for 1981 that is balanced only by reliance on an oil import fee that it may seek to repeal.

The congressional spending blue-print -- aimed at achieving the first deficit-free budget in 12 years -- now goes to a Senate-House conference, which will have to work out differences between the defense-loaded Senate proposal and a more domestic-oriented version adopted earlier by the House.

Before approving its budget proposal by a vote of 68 to 28, the Senate signaled deep trouble for President Carter's oil import fee by deciding overwhelmingly to permit a congressional vote on repealing the fee regardless of whether the action would unbalance the budget.

"In all candor, this is a test vote" on the import fee, said Sen. Bob Dole (R-Kan.), who joined Sen. Howard Metzenbaum (D-Ohio) in advocating what Dole described as a "clear message . . . that the members of the Senate do not want the oil import fee."

The vote on the oil import fee was 75 to 19 -- more than enough to override a presidential veto of a repeal resolution, although the vote did not necessarily reflect how senators would vote on repeal.

The import fee, which is also being challenged in federal court by some House members, will be passed on to consumers starting Thursday in the form of a 10-cent-a-gallon increase in the price of gasoline at the pump.

Carter imposed it as part of his anti-inflation effort to balance the budget in March, and administration officials reportedly lobbied hard against the Metzenbaum-Dole proposal.

Although neither Carter nor Congress has been relying on it to keep the budget deficit-free, many experts are saying that, because of the deepening recession, the budget proposal that is nearing approval in Congress may be in the red without it.

"A vote for the proposed amendment could very well be a vote to unbalance the budget," warned the budget panel's acting chairman, Ernest F. Hollings (D-S.C.), who noted that the budget resolution anticipates that money from the fee will be used for a $10 billion tax cut to spur business expansion and offset 1981 Social Security tax increases.

Not long after, however, the Senate voted to boost proposal outlays for veterans' benefits by $300 million and to cut overall government operating costs to finance a continuation of Saturday mail deliveries.

Together, these actions wiped out the Senate Budget Committee's proposed razor-thin surplus of $100 million and dipped into the anticipated revenues from the oil import fee to the tune of $100 million to keep the budget out of the red.

Both Carter and the two budget committees had avoided tapping the oil import fee to balance the budget, instead reserving the revenues for a possible tax cut or spending cushion.

The vote to increase veterans' benefits was 87 to 6 with Hollings and the Budget Committee's ranking minority member, Sen. Henry Bellmon (r-Okla.), joining the majority to head off an amendment that would have raised veterans' benefits by $500 million. The Senate voted by voice to approve continuation of Saturday mail deliveries after rejecting, 27 to 69, an effort to sidetrack the proposal.

The Senate and House budget committes had voted to eliminate Saturday mail deliveries as an economy measure, although Carter had proposed retaining six-day mail service.

The cutback in civilian operating costs comes on top of a 5 percent across-the-board government reduction in operating costs that the Senate Budget Committee recommended. It means that civilian agencies will be ordered to reduce personnel and other administrative costs by 6 percent. The Pentagon cutback would be left at 5 percent.

The House proposal for administrative savings is far less sweeping, and Sen. Pete V. Domenici (R-N.M.), calling the proposed cutback an "absolute gimmick," predicted that the cuts will never be made. Instead, he said, there will be an overall increase in spending.

The Senate also voted to continue the Law Enforcement Assistance Administration and its program of grants to state and local governments, although no funds would be provided next year. This followed an earlier vote by the Senate to keep the program of revenue sharing with states alive with $700 million for next year, to be achieved by cutting categorical grants to state and local governments. The House cut out funds for LEAA and state revenue sharing.

Although the House-approved budget is also in balance and calls for spending $611.8 billion, only slightly less than the Senate, major differences could produce an "extremely difficult" conference, according to House Budget Committee Chairman Robert N. Giaimo (D-Conn.).

The Senate budget calls for defense spending of $155.7 billion -- a 5.6 percent increase, after adjusting for inflation, over projected outlays for this year. This is $7.8 billion more than the House approved in a defense package of $147.9 billion. The administration proposed $150.5 billion for defense.

To finance its big defense spending increase, the Senate cut a wide, and in some cases deep, swath through existing social programs.

It slashed the nearly $10 billion food stamp program by $1.4 billion, nearly triple the House cutback of $320 million. It anticipated an end to all the 200,000 antirecession public service jobs by the end of 1981, while the House cut the total by 50,000. It provides $1.4 billion less in jobs than the House, $1.3 billion less in transportation, $600 million less in fuel cost assistance, $700 million less in education and $300 million less in health.