National Bank of Washington, the District's third largest bank, reported its best earnings results in two years yesterday as net income in the third quarter increased by more than $1 million.

NBW's dramatic improvement in earnings represents the first real sign of a turnaround at the bank, which has struggled for nearly two years under new management to rid itself of the effects of a scandal involving insider loans.

NBW reported net income of $1.42 million ($1.03 a share) for the third quarter, compared with $272,995 (20 cents) a year ago.

For the first nine months of 1982, NBW had income of $2.57 million ($1.86) before securities transactions, an increase of $745,625, or 41 percent, compared with the same period last year. Net income for the first three months of the year totaled $1.52 million ($1.11), a decrease of $290,402, or 16 percent, compared with $1.82 million ($1.32) last year.

NBW Chairman Luther H. Hodges Jr. said the results represent "dramatic progress" since he was elected chief executive officer nearly two years ago.

Hodges attributed the gains to a decision to restructure the bank's balance sheet, additional steps to control costs, and increases in fee income. Hodges also noted that the third quarter of 1981 included a nonrecurring operating loss of about $250,000 after taxes.

In the second quarter of this year, NBW lost $399,600 because it sold more than $20 million of low-yielding municipal bonds at an after-tax loss of nearly $1 million. At the time, Hodges said the bank could put the proceeds of the municipal bond sales in investments yielding an income level that would make up for the loss within a year.

"Taking that loss in the second quarter was a positive step and has helped in this earnings rebound," Hodges said in an interview yesterday. "Taking that loss was part of the restructuring of the balance sheet."

While the $1.04 per-share earnings in the recent quarter still isn't at a level at which the bank is satisfied, Hodges said, "It is a dramatic rebound compared to the 20 cents per share earned during the same period in 1981."

The earnings are the highest since Hodges was named chairman. Directors elected him chairman in the wake of widespread publicity stemming from irregularities involving insider loans. In the interim, the new management has taken fairly drastic steps to rid NBW of a huge portfolio of low-yielding mortgages and municipal bonds.

Total assets at NBW during the quarter averaged $1.45 billion, compared with $926 million a year ago. Average deposits totaled $802 million in the quarter, compared with $684 million in the first three months last year.