Virginia reached its lowest unemployment rate in three years last month, dropping to 5.1 percent, well below the national average of 9.3 percent, the Virginia Employment Commission said yesterday.

It was the eighth straight month that the unemployment rate fell in the state, as the decline in the labor force more than offset a drop in the number of jobs. The rate in September 1982 was 7.4 percent.

The number of unemployed Virginians fell to 137,500 last month, or 8,000 fewer than in August when the rate was 5.3 percent. Virginia was one of only eight states with an August unemployment rate below 6 percent.

VEC analysts, however, were careful to attribute the rate decline to seasonal factors, people leaving the job market at the end of the summer, especially in categories such as farm laborers, services, construction and the self-employed.

There were, however, gains in local school systems and at food and tobacco processors. Textile, apparel and furniture industries also added workers. Overall, the state's labor force fell by 25,000, while there was a reduction in employment of 17,000 jobs.

"Labor force reductions sometimes occur in September as employment patterns shift from summer to fall," the VEC said. "Students often will seek part-time work in some numbers at mid-September after they are settled into fall classes. Youthful job-seekers did not prove, however, to be a particularly significant factor during the Sept. 11-17 reference week . . . possibly because of early college openings."

Northern Virginia continued to be the healthiest region of the state, with the rate dipping from 3.0 percent in August to 2.9 percent in September. In Richmond, the rate fell from 4.5 to 4.1 percent, and in the Tidewater area it held steady at 4.3 percent.

The highest unemployment rate in the state was 22.3 percent in Buchanan County, located in the state's far southwest, still reeling from the region's depressed coal economy.

About 45,000 Virginians were drawing jobless benefits in September, down from 50,000 in August. New claims for benefits of 3,800 a week last month were down from the previous month's 4,300 a week.