Mayor Walter E. Washington, after 15 days of silence during the confusion and doubt surrounding the Democratic mayoral primary count, conceded victory yesterday to City Council member Marion Barry and pledged his support to Barry in the November general election.

His brief, three-sentence concession statement was issued just hours before the D.C. Board of Elections and Ethics certified the results of the Sept. 12 primary last night.

"I extend to Mr. Barry my personal congratulations, my support in the November general election and my assistance in providing an orderly transition in government," the mayor said.

City Council Chairman Sterling Tucker, who came in second behind Barry in the closely contested race, had conceded victory to Barry on Tuesday and also urged the Democratic Party "to unite in preparation" for the Nov. 7 general election. In that election, Barry feaces Republican candidate Arthur Fletcher, a former assistant secretary of labor in the Nixon administration.

Last night the election board certified the primary election results, which differed only slightly from the last report given Sunday by the board. Final official figures for the top three contenders for the Democratic Party mayoral nomination are:

Barry 32.841

Tucker 31.277

Washington 29.881

In another development yesterday, the D.C. Court of Appeals upheld the refusal Monday of a D.C. Superior Court Judge to block certification of the Democratic primary results in the Ward 5 City Council race.

Robert Artisst, who lost to incumbent William Spaulding by about 300 votes in the Ward 5 race, had sought to block certification, claiming various irregularities in voting procedures in at least three precincts.

Superior Court Judge George H. Revercomb ruled against Artisst, citing D.C. laws specifying that disputes over ballot procedures can be made only after certification and must be brought directly to the D.C. Court of Appeals.

The official results in the Ward 5 race showed Spaulding with 4,343 votes and Artisst with 4,013.

In Ward 6, incumbent Council member Nadine Winter won by 105 votes. She received an official total of 5,046 while Patricia Rice Press got 4,941.

Tucker also had threatened to go to court, in an effort to set aside the entire Democratic primary, but appears to have abandoned that strategy, at least for the moment.

In a press conference last Friday, he had described the problem-plagued election as a "farce" and said he would seek an injunction setting aside the election and ordering a new one.

After Artisst's failure in court Monday, however, Tucker appeared to back away from his position. He said instead he would submit affidavits and other evidence of alleged election irregulaities to elections board general counsel Winfred Mundle, who may refer incidents involing possible fraudulent or other criminal intent to the U.S. Attorney's office.

Tucker said he saw this as a "proper alternative to the legal remedy I had planned to seek." Tucker's attorney, R. Kenneth Mundy, emphasized, however, that this does not preclude Tucker's going to the D.C. Court of Appeals at some later time.