Unemployment fell dramatically in December to a three-year low of 6.4 per cent, the Department of Labor reported yesterday.

The rate in November, originally reported as 6.9 per cent, was 6.7 per cent, the department said.

President Carter called the decline "good news for the country," and took quick credit for a good deal of the improvement.

Carter, at a session with reporters and the chariman of the Council of Economic Advisers, Charles L. Schultze, said the effects of the public works and public service employment programs he backed last year are finally being felt.

Despite the quick self-congratulations at the White House, it is clear that the deep decline in the December unemployment rate came as a surprise to most analysts, including those in the administration.

Only last month presidential spokesman Jody Powell all but wrote off the administration's chances of reaching its goal of driving down unemployment to 6.5 per cent by the end of 1977. "It's not likely to be there," he told reporters then.

Commissioner of Labor Statistics Julius Shiskin said yesterday that the performance both in December and in 1977 as a whole was "truly remarkable." Last month 410,000 people found jobs, according to the Labor Department's survey of a sample of 47,000 households. Over the year; 4.1 million Americans gained employment, a postwar record.

Another Labor Department survey, of employer payrolls - which is less inclusive but more reliable on a month-to-month basis than the household survey - also showed a big pickup in jobs in December and in 1977 as a whole.

The December performance also indicates that earlier unemployment reports - which had shown the jobless rate stuck between 6.9 and 7.1 per cent from April through November - probably masked what has been a steady improvement in the nation's economy throughout the year, Shiskin told the congressional Joint Economic Committee.

The problem is with the adjustment the department makes in the unemployment rate each month to account for seasonal ups and downs in employment and joblessness. When the department revised its adjustment formula, as it does every year, it found unemployment had declined gradually thoughout the year, rather then being stuck on a plateau through November, then falling sharply.

The dip in unemployment should strengthen the President's hand in pressing Congress to hold down spending, and it may be an indicator that the economy will be stronger in 1978 then originally anticipated.

At the meeting with the President, economic adviser Schultze was quick to point out that the economy still needs the stimulus the President's expected $25 billion tax-cut package would provide, starting Oct. 1.

Shiskin told legislators yesterday that if most economic forecasters are correct and the economy has another good growth year in 1978, the unemployment rate could be below 6 per cent by year's end.

Courtenay M. Slater, chief economist for the Department of Commerce, was slightly less optimistic, telling legislators that unemployment should be near, but not necessarily below, 6 per cent by year's end.

She also predicted that statistics the Commerce Department releases next week will show the economy grew at a 4 per cent rate or less in the final three months of 1977, a pace slower than most economists estimated and probably not consistent with the strong increase in jobs during the last quarter.

The improvements in December were across-the-board and affected nearly every group the Labor Department surveys. While 410,000 people founds jobs, the number of persons in the labor force - those with jobs and those actively seeking them - fell slightly, by about 70,000.

But department economists noted that the labor force grew by nearly 900,000 in November, a huge increase, and that such big jumps are often followed by small rises or samll declines. Employment grew by more then 900,000 in November.

The unemployment, rate, which is based on a survey of 47,000 households, is the percentage of the labor force unemployed but actively seeking work.

As the overall rate fell to 6.4 per cent - it was 7.8 per cent a year ago - the rate for adult men fell from 4.9 per cent in November to 4.7 per cent, while the rate for adult women fell from 7.1 to 6.7 per cent. The teenage unemployment rate declined to 15.4 per cent from 17.1 per cent, and the black unemployment rate fell to 12.5 per cent from 13.8 per cent. Black teenage unemployment, which was 39 per cent in November, was 37.3 per cent in December.

Over the year, the number of unemployed persons declined by 1.1 million, to 6.3 million, while the labor force grew by nearly 3 million - far in excess of the usual growth of 2 million - to 98.9 million.

The survey of employer payrolls showed a growth of 215,000 jobs in December and 3.1 million over the year. The department said that were it not for the coal strike, there would have been a gain of 365,000 workers last month. While strikers do not show up on payrolls, they are counted as employed in the household survey.

The Labor Department said there was a gain of 160,000 jobs in manufacturing, 20,000 in construction and 40,000 in transportation equipment - which reflects the end of the Lockheed and Boeing strikes.

The Labor Department also reported that the number of workers who would be looking for work but felt they would not be able to find any - the so-called discouraged workers - fell from an average of 1.1 million in the middle of the yearto 970,000 during the last three months.