The House authorized an additional $22 million yesterday to complete the process of cleaning up the financial confusion that has made it impossible to audit the books of the District of Columbia government.

Implementation of a modern financial management system, scheduled to be completed Oct. 1 at a total cost of $38 million, will make it possible for professional auditors to make sense out of the city's finances for the first time.

The city and federal governments are sharing the cost of the two-year project, although Rep. Millicent Fenwick (R-N.J.) said before yesterday's voice vote that the U.S. ought to pay all of the costs because it controlled the D.C. government during the years when the problem developed.

A bill nearly identical to the one approved by the House yesterday is scheduled to be voted on by the Senate next week.

If the project is completed on schedule, the first independent audit of the District of Columbia's finances will examine transactions recorded under the new system for the fiscal year ending Sept. 30, 1980.

With a certified audit in hand, D.C. auditor Mathew Watson said the city government will be able to go to the commercial bond market to borrow funds, a situation that he said will save the city several million dollars a year in interest.

The city has been forced to borrow money over the year from the U.S. Treasury, where the interest rate has been about 3 percent higher than on the commercial market, Watson said.

The question of borrowing by selling tax-free municipal bonds on the commercial market did not come up until home rule legislation, passed by Congress in 1971, made the option possible.

When the first attempt to audit the city's books was made, in 1975, the firm of Arthur Andersen & Co. reported that records were in such bad shape, largely as a result of years of uncentralized Congressional control, that they could not be audited.

As a result, Sen. Thomas Eagleton (D-Mo.), chairman of the Senate District Committee, introduced legislation that created the Temporary Commission for the Financial Oversight for the District, which is directing the project.

Bruce Rohrbacher, executive Director of the commission, said yesterday that most of the money is being spent on contracts awarded by competitive bid to private accounting and management firms.