Weary from pervious battles with the Senate over funds for abortions, the House yesterday afternoon relented from its hard-line stance on the issue and passed the District of Columbia's $1.4 billion fiscal 1980 operating budget.

By voice vote, the house reversed its earlier support for an amendment suponsored by Rep. Robert K. Dornan (R-Calif.). That amendment, which had been rejected by the Senate, would have prohibited the District government from spending even its own tax money on abortions for the poor.

The House instead adopted the same weker antiabortion amendment that Congress accepted last Friday as part of a continuing resolution to keep federal agencies functioning. That language would prohibit the use of federal -- not city -- funds to pay for abortions except in cases where the mother's life is endangered or in promptly reported cases of rape or incest.

The issue now goes back to the Senate, which had adopted the same language by a single vote in a compromise last Friday.

A conference committee had agreed on all other aspects of the District budget, but until both chambers agree on abortion funding, the city cannot begin new or expanded programs envisioned in the budget for the 1980 fiscal year, which began Oct. 1. Along with many federal agencies, including the Defense, Agriculture, Labor and Health, Education and Welfare departments, the District government is now operating under terms of the continuing resolution adopted last Friday, which allows spending to continue at fiscal 1979 levels.

Rep. Charles Wilson (D-Tex.), chairman of the House District appropriations subcommittee, said acceptance of the ydornan amendment, the most restrictive abortion language yet approved by either chamber, would have been "catastrophic" for the city's poor. with that restriction removed, the city can now fund abortions with locally raised tax money, as can all other state and local governments in the country.

Dornan, who opposes abortion for any reason, said he offered his most restrictive amendment for the city after learning that abortions outnumber live births by nearly 2-to-1 in the district.

The overall budget includes a federal payment of $238.4 million, which is considerably less than the $315 million the city government sought as compensation for the number of nontaxable federal installations in the city.

Rep. Wilson said yesterday's compromise on the abortion amendment was worked out in advance with the cooperation of "some key Republicans."

Among Republican members cooperating were yrep. Henry J. Hyde of Illinois, the author of the 1977 amendment that first prohibited the expenditure of federal funds for abortions.

Other Republicans who helped "slide it through," in Wilson's words, were Reps. Silvio O. Conte of Massachusetts, Carl D. Pursell of Michigan and Robert Bauman of Maryland.

Conte helped by agreeing with House Speaker Thomas P. (Tip) O'Neill Jr. over the weekend to go along with the compromise; Pursell aided the cause by "not giving Dornan enough time to make a fuss," according to Wilson; and Bauman, a conservative parliamentary tactician, helped by doing nothing.

Speaker O'Neill greased the skids by interrupting consideration of the Department of Energy's budget to allow Wilson to call up the D.C. budget before opposition forces could organize.

The Republicans managed to defeat, again by voice vote, a move by Rep. Louis Stokes (D-Ohio) to get the House to go along with the Senate's version of the city budget and exclude any prohibition on abortion funding.

Wilson credited the action to a combination of "not advertising that we were bringing it up" and a general feeling by those who were on the floor, numbering perhaps 100 of tthe 435 members, that "we didn't want this to become another big battle." Wilson cited "real good strategy planning by the staff" of his subcommittee in getting the compromise approved.

Still to come is floor action by the Senate. Wilson said supporters "will have to plan it carefully there, too."

While the latest abortion language made it through the Senate by the narrowest of margins, the chances that the Senate will accept it this time around appear better.

The Senate's conferees on the D.C. budger, led by Sens. Patrick J. Leahy (D-Vt.) and Charles McC. Mathias (R-Md.), had opposed any restictions on abortions in the D.C. budget while House conferees had supported the Dornan amendment.