The first full audit of the D.C. government's finances confirmed yesterday that the city rolled up a $105 million deficit last year and approached insolvency as its cash reserves dwindled and unpaid bills mounted.

But for all the grim fiscal news it contained, the issuance of the audit represented a positive achievement for the city, because it opens the way for an approach to Congress and private lenders for help in paying off the city's staggering debts. For the first time, District officials are equipped with numbers not open to challenge by skeptics as they search for loans and legislation to bail out a floundering city government.

The audit, a 100-page report that represents the first unequivocal, independent analysis of the city's finances ever done, confirmed that the city has been living beyond its means for years. But the impact of the grim figures was softened by the fact that the city is actually better off than Mayor Marion Barry and his aides had predicted. The $105 million operating deficit for the 1980 fiscal year was less than the $125 million Barry had predicted. The cumulative deficit, including the previous 10 years, is $388 million, not the $409 million previously estimated, the audit showed.

On Sept. 30, the last day of the 1980 fiscal year, the city had only $9.3 million in operating cash and owed more than $140 million to suppliers, including the federal government. City Administrator Elijah Rogers said at a press conference that the city government used revenues from the current fiscal year to pay those bills, leaving the District with a "serious cash flow problem." He estimated that the city would have to borrow $184 million in cash before the end of September to keep meeting payrolls and other current obligations. That figure is also an improvement over previous estimates, which called for borrowing $215 million.

Barry and Rogers have spent many months warning the public and Congress what the audit would show. The actual figures allowed the mayor to claim, as he did in a statement yesterday, that his cost-cutting and payroll-paring measures, "while painful, were the approach we had to take to resolve the problem."

The deficit was the greatest in the city's modern history and the overspending violation of congressionally imposed spending limits. But Barry and his aides chose to see the issuance of the audit and the improvement over earlier predictions as positive developments.

In fact, the mayor, whose energies have been consumed for the past year by his effort to grapple with the city's financial problems, used the occasion of the publication of the audit to proclaim a whole new approach to running the city. He did to the financial crisis what wits of the 1960s used to say the United States should do about Vietnam: he declared victory and got out.

After a painful year of financial scrambling in which he personally presided over details of budget-making and issued his own plan for paying off the deficit, he said yesterday that "as the city's financial management and financial position improve, I am now able to devote more energy to improving District government services and programs. I am particularly anxious to focus attention on my high priority program areas of housing, crime reductions, jobs and assistance for youth and the elderly."

Because the budget deficit for 1980 had been expected, the audit contained no major surprises. Its significance lies less in the numbers than in the fact that it was issued at all. It is the culmination of a five-year effort to produce an independent audit of the city's books, which Congress determined in 1976 to be so chaotic as to be unauditable.

The issuance of a "clear" or "unqualified" audit by independent accountants is a prerequisite to the District's entry into the commerical bond market, a keystone of the city's long-range financial strategy. With that step completed, Barry and other city officials can now concentrate on a program for paying off the accumulated deficit.

The mayor issued a plan for doing that last summer, but at least three other plans have been offered since then -- by City Council Chairman Arrington Dixon, City Council Finance and Revenue Chairman John J. Wilson (D-Ward 2) and D.C. Del. Walter E. Fauntroy. Barry's plan proposed borrowing to pay off the deficit from the federal Financing Bank, an arm of the Treasury. Fauntroy and Dixon emphasized borrowing from banks. Wilson said the money should be raised through tax increases.

All have been huddling since last week in a search for a program they could agree upon, and Rogers said he expects an announcement late this week of "a D.C. position supported by all the key actors." He said "we will ask Mr. Fauntroy to introduce a bill to authorize the city to pursue several options."

The audit was prepared by the accounting firms of Arthur Andersen & Co. and Lucas, Tucker & Co. The task took two years and cost $6.1 million, a sum far higher than similar undertakings have cost in other cities. Robert Stephens, executive director of the Financial Oversight Commission, the body set up by Congress to improve the city's financial management practices, said the actual cost of the audit is $1 million. The rest, he said, was for "tasks that the [city] would normally have performed for itself" -- that is, for staff work that District employes were unable to perform on their own.

As expected, the biggest opponent of last year's $105 million deficit was overspending by the city's mammoth Department of Human Services. Authorized to spend $290.5 million, the department actually spent $323.6 million, an excess that city officials have attributed to obligatory programs such as welfare and food stamps that they say were underbudgeted to begin with. s

"We are aware of the program at DHS," Rogers said. "We are on top of it." James R. Boyle, who was dismissed last month as finance director of the D.C. public school system, has been named comptroller of DHS.