Unemployment in Virginia rose to a new post-World War II high of 8.4 percent last month, the Virginia Employment Commission reported yesterday.

The jobless level in January on the same unadjusted basis was 7.8 percent, which, until the February figures were reported, was the highest since the war. The sharp jump in unemployment last month was because of layoffs in the auto component, furniture and textile industries, the VEC said.

In addition, there was a rise in the number of people looking for work, and who were therefore counted in the labor force, VEC analyst William Mezger said. There were 220,200 Virginians out of work in February, compared with 203,200 in January, he reported. Meanwhile, the labor force expanded by 16,600 people during the month, a large increase for the time of year.

The comparable national rate of unemployment in February was 9.6 percent. After seasonal adjustment, the national rate was 8.8 percent. Most economists expect the underlying jobless level nationwide to rise even further in coming months. However, as the weather improves there is usually a seasonal decline in the numbers out of work in the spring and summer months. The VEC predicts that the jobless rate for March will stay at about the February level.

Unemployment varies widely across the state, although there were increases last month in five of the six major urban areas. Yesterday's figures showed:

* Richmond unemployment rose from 5.4 percent in January to 6.1 percent in February, a post-war high.

* In Newport News-Hampton, unemployment increased from 7.4 percent to 7.8 percent.

* Northern Virginia, still the lowest unemployment area of the state, saw an increase from 4.5 percent in January to 4.8 percent last month.

* In Roanoke, the jobless level went from 7.6 percent to 8.2 percent.

In contrast to these increases, unemployment in the Norfolk-Virginia Beach-Portsmouth area fell from 8.4 percent in January to 8.2 percent in February, the VEC said.

Of the state's eight metropolitan areas, the Petersburg-Colonial Heights-Hopewell area had the highest unemployment rate--10.9 percent in February.

In February, 76,000 Virginians drew unemployment benefits, the VEC said.