Rep. Harold Washington (D-Ill.), the first black to be elected mayor of Chicago, moved swiftly today to begin healing the intense political and racial strains caused by his bitterly fought mayoral battle with Republican Bernard E. Epton.

Washington called together most of his political antagonists, including outgoing Mayor Jane M. Byrne and Cook County State's Attorney Richard M. Daley, and the hierarchy of the city's clergy to a private "unity luncheon" at the Conrad Hilton Hotel.

Despite his promise to attend, Epton, a wealthy lawyer, skipped the luncheon to fly to Palm Beach for a rest. However, he sent his brother, retired circuit court judge Saul Epton, as an emissary to deliver a private message to Washington, 60.

Washington later told reporters that it was "a clear-cut good-will message," which is a far cry from Epton's sour remarks late Tuesday night when he realized he would lose. Then, Epton told a local television reporter that he hoped Washington would "learn to pay his bills promptly and certainly pay his taxes promptly."

With 99 percent of the city's 2,914 precincts counted, Washington had 656,727 votes, or 51.4 percent, to Epton's 617,159, or 48.3 percent. It was the closest mayoral election here in 80 years and the most racially split. The record-breaking turnout of about 88 percent also set a mark for blacks and whites voting on racial lines.

The racial and political stresses caused by the slashing personal attacks that marked the campaign have made city leaders and outsiders worry about racial harmony under the new mayor.

Major city departments, such as the police, have been dominated by whites throughout the city's history. The police department is one of the most politicized in the nation.

Police Superintendent Richard Brzeczek made television ads supporting Byrne against Washington in the bruising Feb. 22 primary and then announced that he would resign the day Washington takes office.

As he did early today in his victory speech before thousands of ecstatic supporters at Donnelley Hall, Washington again sought to assuage the concerns of white voters who overwhelmingly voted for Epton.

Flanked by nearly 30 civic leaders and clergymen, including Cardinal Joseph L. Bernardin, at a news conference after the unity meal, Washington said: "As the mayor of this great city, it will be my responsibility to reach out to every neighborhood. Over the next four years, there will be total and complete outreach into every area of this city."

He said that "for the first time in our history, a Latino and a black" will have an equal opportunity with a white to be chosen as superintendent of the 16,000-man police force.

And he reaffirmed a campaign pledge that the next police chief "will be from the command structure of the police department." In a city that is more than 40 percent black, the police force is 24 percent black.

Washington's swearing-in date, to be set by the City Council, probably will be late this month or in early May, after yesterday's returns are officially certified. The 50-member, all-Democratic council met today, paying tribute to Byrne and the 18 aldermen who are departing--some, including the mayor, into a retirement forced by the voters. Because of victories by independents and other turnovers, Washington will begin his four-year term at the head of America's second-largest city with a coalition of 16 black aldermen and four white independents.

During the campaign, he excoriated the handful of aldermen, such as former congressman Roman Pucinski, who defected and backed Epton. Today's council meeting was marked by an air of expectancy as to how the new mayor will deal with them and the council. Pucinski was making conciliatory gestures.

The city provides 41,000 municipal jobs and this year's budget is $1.9 billion, both nominally controlled by the mayor. But under the Democratic machine, control was wielded by the Democratic committeemen in each of the 50 wards, putting their supporters on the payroll, firing renegades and extracting generous "contributions" to the party coffers in return. Washington has vowed to end this.

But first he may have to deal with major financial problems. Edwin C. Berry, chairman of Washington's 43-member transition committee, said there will be a sizable deficit in the 1983 budget, and the school system is $100 million in the red. The Chicago Transit Authority is nearly bankrupt, Berry said, and the federal government is pressing Chicago to repay millions of dollars in community development grants.

Berry said "there will be no major shakeup to benefit minorities" in the police force or other agencies. "There will be no hiring in a wholesale way or replacement of jobholders with blacks or Hispanics. But there will be an effort to see that these groups are properly represented at all levels of government as we move forward, through attrition."

In record numbers Tuesday that are expected to exceed 1.3 million overall, whites voted for whites and blacks for blacks. But exit polls and voter profiles also show that Washington did well with Hispanics and better than expected with whites.

His strategy aimed at turning out the highest possible black vote in the hope of overcoming a white backlash that gathered momentum as ethnic Democratic leaders defected to Epton or sat on their hands.

Washington concentrated on the 19 west and south-side black wards he won and which gave him a 36.3 percent plurality in the Feb. 22 primary, when the city's whites split their votes between Byrne and state's attorney Daley.

Yesterday, he easily won those wards. Six of them gave him a total of 158,641 votes or 97 percent, while Epton got only 5,665 votes, 3 percent. Washington added Hyde Park and two other wards, including the Hispanic 22nd, which had gone to Byrne or Daley earlier.

Epton's strength peaked in the Byrne-Daley areas, where in four wards on the west and north west, he rolled up 96 percent of the vote. His margins were smaller in the other 24 wards he won, especially in three west-side Hispanic wards, where his margin totaled just 3,626. Both candidates courted Hispanic voters with promises of a wider City Hall role for them.

Epton won the six lake-front wards, seat of Chicago's independent and reform movements for the past 20 years. But Washington cut heavily into this, and Epton won them by only 57 percent overall.

Analysts say the large undecided vote in these wards swung heavily to Washington over the weekend partly because Washington, on the defensive through most of the seven-week campaign about his legal problems, went on the attack in the final week, charging that Epton was a puppet of the Reagan White House.

Epton hurt himself with this block of voters by refusing at the last minute to appear on NBC's "Meet the Press" television program last Sunday and by slacking off his campaign schedule in the last few days. While Washington hustled for votes Tuesday, Epton went to the home opener of the White Sox.