The Virginia Electric and Power Co. won a rate increase of $59.9 million yesterday, enough to boost the bill of the company's typical residential customer $2.58 a month.

"The company certainly is not jubilant about it," said Paul G. Edwards, a Vepco spokesman who noted that the company had sought a much larger increase. Edwards predicted, nonetheless, that customers will pay smaller bills in the coming months if the utility can keep its nuclear power plants operating.

The Richmond-based utility had asked the Virginia State Corporation Commission for an increase that would have totaled $72.6 million and added $3 a month to the average residential electric bill. The company's request was opposed by consumer groups and the state government's advocates, but won the approval of a hearing examiner and two of the three men who run the agency.

The commission told Vepco late yesterday afternoon the new rates could go into effect Oct. 1 and directed the firm, which serves Washington's Virginia suburbs and most of the state's major population centers, to submit new schedules reflecting the higher rates before that date.

Utility executives defended the increase, saying the company urgently needed the higher revenues to offset inflation and they predicted activation of the firm's four nuclear plants should produce a decrease of 7.2 percent in electric bills by the end of 1980.

Commission Chairman Preston C. Shannon and Commissioner Junie L. Bradshaw concurred with the recommendation of hearing examiner Stewart E. Farrar that the increase be set at $59.9 million. But Commissioner Thomas P. Harwood, Jr. dissented, saying he would have set the increase at a lower figure -- $52.26 million. Harwood said the agency should have recognized that Vepco would pay lower taxes because of a change in the way it accounts for its interest on borrowed funds. ". . . I dissent from the majority's failure to make that adjustment in this case," he said.

The last increase granted Vepco on April 1 totaled $18.1 million. When the latest application for higher rates closely followed, there was a wave of protests from consumer groups who said the utility already has one of the highest residential electric rates in the country. "We're a little astounded by their audacity in coming in for yet another," complained Jerri Brown, president of the Virginia Consumers Conress.

A typical customer who used 1,000 kilowatts hours during January would pay $62.48, Vepco spokesman Edwards said. By next January that same amount of electricity will cost $57.98, assuming all four of the company's nuclear units are operating on schedule, he said.

Vepco has had difficulty getting all of the nuclear units into simultaneous operation, which has forced it to rely on more costly fossil fuel power plants. Since customers pay for the more expensive fuel plus interest for

The size of yesterday's award was a setback for State Attorney General J. Marshall Coleman and various consumer groups which had opposed it.

The company's management, which has undergone major changes in the past year, has been criticized as being unable to cope with both changing regulatory environment and sharp decreases in the rate of growth in electricity consumption.

North Carolina now is pressing the Federal Energy Regulatory Commission to investigate the company's management and finances, citing the lower costs of power from other utilities in the state.