In an unusually harmonious atmosphere, the Virginia House of Delegates yesterday approved a two-year state budget that would give Northern Virginia $15.5 million more than former Gov. Mills E. Godwin had proposed when he sent the budget to the General Assembly.

The chief beneficiaries of the extra spending in Northern Virginia would be the school systems of Arlington ($3.8 million). Alexandria ($2.6 million) and Fairfax County ($2.2 million). The Metro subway would get another $4 million (for a total of $10 million) and the Northern Virginia Training Center for mentally retarded would get about $1 million in new funds.

Without the added funds, the local governments would have to find other ways to raise the money -- possibly by raising local taxes.

The budget approved by the House and sent to the Senate provides for total spending of about $9.2 billion from July 1, 1978, to June 30, 1980 -- about 18 percent more than the state is expected to expend in the present budget period, ending June 30.

The House-approved budget proposes $76 million more in spending than had been recommended by Godwin. The bulk of it -- $51.6 million --would go for a 4.8 percent cost-of-living pay increase for state employes in 1979, the second they would get during the biennium. Godwin's proposal already called for a pay boost this year.

The extra spending would not require a tax increase, according to House Appropriations Committee estimates. It would be financed chiefly from the continuing improvement in state revenues forecast by Gov. John N. Dalton -- a trend signaled earlier by Godwin -- and by deferring payment of some of the money the state owes its highway construction fund.

The House took up the budget yesterday in an atmosphere marked by little dissension or grumbling from delegates who in the past have wanted more money for pet projects in their district or sought slashes in government spending.

Part of the reason was that with Virginia's, and the nation's, economy rebounding from the doldrums, state revenues appear sufficient to keep pace with expenses, and even provide a cushion for some extra spending beyond normal growth.

But many delegates gave credit to the Appropriations Committee, which, under its new chairman, Richard M. Bagley (D-Hampton), held all its budget-making meetings in the open for the first time. Part of the openness was dictated by extensions of the state's Freedom of Information Act, but part of it was the result of Bagley's personal style.

On his own, for example, he informed the Republican caucus of amendments calling for additional spending.

After the House adopted the Appropriations Committee's budget, Del. Raymond E. Vickery Jr. (D-Fairfax), in the past a frequent critic of the House's more conservative leadership, took the floor to praise Bagley, a moderate-conservative from urban Tidewater.

"In past sessions," Vickery said, "questions have been raised about the openness and thoroughness (of the budget process). I can say, without qualification, that this budget was arrived at in the most open way possible. I want to show my appreciation, and I am sure the appreciation of this House, for the magnificent job done by the chairman of this committee."

The budget bill was approved swiftly by a 96-to-1 vote -- the dissent coming from Del. Robert L. Thoburn (R-Fairfax), who wanted a proviso that "no funds shall be used to kill unborn humans." The reference was to the House's earlier passage of a bill that would overturn a Dalton decision and permit abortions for indigent women to be paid for by the state.

But Thoburn's proposed floor amendment -- the only one offered --was killed with a thundering chorus of noes.

Northern Virginia delegates were generally pleased with the amended budget bill. In a joint statement, three Fairfax delegates, Dorothy S. McDairmid (D), Warren E. Barry (R) and Vincent F. Callahan Jr. (R), said, "Northern Virginia fared exceptionally well . . . with proposed appropriations remaining intact and new money added for elementary and secondary education, higher education, mass transit, mental health and corrections."