The last die-hard warriors in Lyndon Johnson's War on Poverty are working overtime these dwindling summer days on a final, bitter mission: to carry out the death sentence for their own agency, the Community Services Administration, by Sept. 30.

It is the first time a federal bureaucracy of significant size has been extinguished, officials say, since World War II.

With the passing of the CSA, whose programs will be dispersed to the states, the nation's poor lose their official champion in Washington, albeit a champion regularly condemned as wasteful and incompetent.

When the poverty fighters launched their heady crusade in 1964, their stated intention was nothing less than to "win," to eliminate most of the nation's poverty by around 1976. The head of the new Office of Economic Opportunity (OEO) was the ebullient Sargent Shriver, brother-in-law to the late President Kennedy, popular head of the Peace Corps, a man with ready access to President Johnson.

But the poverty rate has changed little since those brave beginnings and the political climate has altered dramatically. The hyperbole this season at CSA, the last surviving remnant of OEO, tends toward resignation, even cynicism.

"We are asking our people here to take on the toughest task any group of federal employes has had during peacetime," said CSA head Dwight Ink, a 30-year veteran of the federal bureaucracy.

President Reagan last spring appointed him head of the agency for the sole purpose of killing it. "We have trouble finding a precedent for what we're doing."

Thus, CSA's 1,000 employes are reluctant pioneers in the field of dismantling an agency. In these final days, the CSA is still responsible for administering grants of millions of dollars to 2,000 community action groups and other grant recipients.

At the same time, its employes are hurrying to acquaint state officials with their new responsibilities before the baton is passed, and struggling with the monumental red tape connected with eliminating a bureaucracy.

They "have a far greater workload than usual at a time when morale is very low and they are also having to scramble for jobs," Ink said. "I admire some of these career people . . . who've believed in the mission so much they stayed on despite a hostile political environment."

The antipoverty agency has spent billions of dollars over the years in a variety of programs, from job training for young people, to health care, to meals for the elderly, to bus service in Appalachia, all designed to help the poor become economically self-sufficient.

The Nixon administration attempted to eliminate the agency but was blocked by a court decision. Instead, OEO was whittled down to CSA.

Now the Reagan administration, as part of its move to shift more authority from the federal government, has won congressional approval to send funds targeted for poor people directly to the states, where the decisions on how to spend them will fall to the governors.

But Congress refused to give the states total discretion, as Reagan wanted, and insisted the money be earmarked for antipoverty programs.

Rather than transferring the duties and personnel of CSA into some larger department, the traditional game of bureaucratic musical chairs, this entity is truly doomed, Ink said. The only remaining trace of a federal antipoverty agency will be a small Office of Community Affairs at the Department of Health and Human Services, where grants to the states will be administered.

Critics fear some states will be indifferent custodians of programs for the poor, as some have been in the past. Ink acknowledges there will be "variations" in effectiveness from one state to another, but defends the move. The federal role was always intended to be a temporary one, he said.

Besides, the environment of the 1980s includes elements which did not exist when federal intervention was deemed necessary: an increase by the states in their share of spending for social programs, better organized community groups and heightened visibility given the problems of the poor by the media and others.

Still, some veteran antipoverty workers at CSA disagree passionately as they stare at their federal pink slips.

"It's a joke," said John Macomber, who has worked for the agency throughout its 17-year existence. Like many other CSA employes, his tone is a mix of disillusionment and pride, anger and resignation, as he talks about the "excitement and challenge" of the early days, about what went wrong, and traces such derailments as the war in Vietnam and the intransigence of problems at home.

"We were going to eliminate poverty by 1976," he said, fiddling nervously with a drawer of his desk. "Now we're going down the drain. I feel very sad. Sad for the people this agency and programs have served. They will bear the brunt of the burden."