Sales of existing single-family homes rose by 15 1/2 percent in January, in the largest one-month increase in the 16 years that such records have been kept, the National Association of Realtors said yesterday.

Existing houses sold at an annualized rate of 2.61 million in January, the real estate trade association said.

"The record improvement . . . is clear-cut evidence of the positive impact lower mortgage interest rates are having on the housing industry," said Jack Carlson, chief economist and executive vice president of the association.

The association's president, Harley W. Snyder, said that "thousands and thousands of people are coming back into the marketplace as a result of lower interest rates, and they are discovering that the home they contemplated purchasing last year but couldn't afford because of high interest rates is worth another look now."

Harry Pryde, president of the National Association of Home Builders, said "the housing recovery is gaining momentum. As market conditions improve, people who don't have to move are starting to upgrade their housing."

Both Carlson and Pryde warned that the strong market can continue only so long as interest rates stay in or below their current range.

"There's plenty of room for a further reduction in interest rates," Pryde said. "There is no justification for the more-than-seven-point spread between the inflation rate, now under 5 percent, and mortgage rates, which remain in the 12 to 13 percent range" on the average.

Added Carlson: "It is essential that the administration and Congress get control over expanding deficits so that interest rates can continue on their downward path and the national housing recovery can continue."

Mortgage interest rates, which touched 18 percent in some areas in the past two years, began declining sharply in August. They have leveled off since then, and forecasts are mixed for the future, although most economists expect them to remain in their current range at least into the summer.

The actual number of resales in January was 157,000, compared with 148,000 in December and 116,000 in January 1982.