Recession-squeezed companies slashed payrolls in the third quarter but still managed to produce more goods and services, a combination of events that sent productivity climbing at a healthy 3.6 percent seasonally adjusted annual rate, the Labor Department reported yesterday.

That increase for nonfarm businesses was the result of a 0.7 percent rise in output and 2.8 percent drop in hours worked, both at an annual rate.

Compensation per hour, which includes employer-provided fringe benefits and social insurance payments as well as wages, climbed at a 6.6 percent rate. With the productivity gain, union labor costs -- a major determinant of inflation--went up at only a 2.9 percent rate, the smallest increase since the first quarter of 1977, the department said.

The department also indicated in another report that payrolls still are being reduced. Initial claims for regular state unemployment benefits rose to a seasonally adjusted 687,000 in the week ended Oct. 16, a 3,000 increase over the previous week and the fifth week in a row that the figure has hovered near the 700,000 level, the department said.

The average level of claims for the five weeks is by far the highest for new claims for any period since the recession began in July 1981. According to labor market experts, this high level of claims means that the number of jobs nationwide is falling by about 200,000 a month.

The latest information on unemployment claims also means that the unemployment rate, which was 10.1 percent in September, likely will climb to 10.3 percent or more for October, the analysts said. The October unemployment rate will be reported next Friday.

Productivity at nonfarm businesses last rose as much as it did in the July-September period in the first quarter of 1981, when it increased at a strong 4.9 percent annual rate. That increase, however, was the result of a surge in output rather than a cut in hours worked.

The third-quarter gain in productivity was nearly twice as large and was generally among nonfarm businesses. Output per hour worked in manufacturing jumped at a 7.1 percent rate as hours tumbled at an 8.5 percent rate and output went down at only a 1.9 percent rate. It was the largest gain in manufacturing productivity since the fourth quarter of 1980, when it shot up at a 14 percent rate.

Manufacturers of nondurable goods managed to increase output at a 3.3 percent rate while reducing the number of hours worked at a 4.3 percent rate, giving them a 7.9 percent rate of increase in productivity.

Durable-goods makers had a smaller but still sizable increase in productivity, which rose at a 6.4 percent rate. Their output fell at a 5.5 percent rate but the number of hours worked dropped much faster, at an 11.2 percent rate.

When figures for agriculture are included along with those for nonfarm business, productivity for the entire private business sector rose at an annual rate of 4 percent for the quarter.