The Virginia Senate today approved a $13.1 billion budget that deliberately fails to spell out how the state will spend its transportation money over the next two years.

The $1.7 billion in "undesignated" transportation funds was never mentioned during the budget debate, even though many legislators regard it as the single most pressing issue to be resolved before the General Assembly's scheduled Saturday adjournment.

Under an elaborate and elusive strategy worked out last weekend by the Senate Finance Committee, Democratic leaders -- including Gov. Charles S. Robb -- have chosen to leave the hotly contested decisions on transportation funds to the six members of a joint budget conference committee who are expected to meet privately during the next two days.

Northern Virginia legislators are hoping that these meetings will produce additional money for the Washington area's Metro transit system -- perhaps the most major unsettled issue in the Assembly's budget wars. They fear, however, that even the presence of Sen. Adelard Brault (D-Fairfax) on the budget conference committee will not be enough to win them the state funds they need to qualify for continued federal Metro construction money.

A second conference committee is meeting this week to decide the fate of a proposed increase in the state's 11-cent gasoline tax -- funds from which are already written into the two-year state budget passed by a 36-to-2 vote in the Senate today.

The Senate added $94.6 million to the budget recommendations from the House two weeks ago, including a $3,000 raise for each of the 140 legislators that would go into effect in 1984. Virginia General Assembly members are now paid $8,000 a year plus expenses.

That extra $94.6 million was made available after the legislature recently revised the state's corporate tax laws. The revisions, sought by Robb late in the session, defer for two years tax credits for large corporations triggered by President Reagan's federal tax cuts.

The bulk of the additional money -- or $30 million -- is earmarked for teachers' salaries. Robb has pledged the funds as the state's share of 10 percent salary increases for teachers in each of the next two years. Although the House already had shifted approximately $53 million for teachers pay raises, that amount was considered insufficient to meet Robb's goal.

Another $11 million went to faculty salaries at state colleges and universities. "We are concerned . . . that many of our higher education faculty trail far behind their peers in comparable institutions," said Sen. Hunter B. Andrews (D-Hampton), acting chairman of the Senate Finance Committee.

Using the new corporate tax revenues, the Senate put $12 million toward building a new medium-security prison in Augusta County. Construction of that jail had been postponed by both former governor John N. Dalton and Robb.

The budget also contains $500,000 to finance legal aid for the poor, a program that recently suffered a 25 percent cut in federal funds. The budget item would give the state's 14 legal aid societies far less than the $3 million they had sought in separate legislation. That legislation, which would have raised court filing fees to pay for the program, is still before the Senate Finance Committee.