With sales at its shopping centers up 10 percent in the first nine months of 1981, Rouse Co. of Columbia reported yesterday a strong gain in operating earnings for the third quarter and year-to-date. Separately, Washington Real Estate Investment Trust earnings continued at record levels during the third quarter of 1981.

Rouse, which recently announced an agreement for Canadian investors, Trizec Corp. Ltd., to buy a one-fifth interest in the Columbia company, said third-quarter earnings before noncash charges (depreciation and amortization of property as well as deferred income taxes) rose to $4 million from $3.4 million a year ago, a gain of 18 percent. Nine-month earnings before noncash charges rose 14 percent to $11 million from $9.7 million.

The real estate development and shopping center firm relies on earnings before noncash charges as the best indicator of performance. Net income for the recent quarter was $495,000 (3 cents a share) compared with $1.05 million (8 cents) a year earlier.

Four new shopping centers were opened or expanded in the recent quarter--White Marsh in northeastern Baltimore County; North Star Mall in San Antonio, Tex.; Salem Mall in Dayton, Ohio, and Columbia Mall, where a Sears store and 70 smaller units were added.

Rouse said all operating divisions posted higher earnings in the 1981 period. An agreement with Trizec of Calgary, is expected to be completed by year's end, at which time Trizec will own 20 1/2 percent of Rouse common shares in exchange for a $36 million investment.

WRIT, of Bethesda, said profits rose 39 percent to $1.14 million (23 cents a share) in the July-September period compared with $817,840 (18 cents) a year earlier. For the first nine months, WRIT profits were $3.2 million (68 cents) versus $2.45 million (54 cents). Per-share figures in both periods reflect a 3-for-1 stock split last March 16.

A company announcement attributed the gains to the more than doubling of rents on July 1 at a WRIT-owned office building, 1901 Pennsylvania Ave. NW; larger apartment rent increases in trust-owned buildings and interest income from a recent offering while WRIT searches for new real estate investments.

Total revenues in the nine months were $10 million compared with $9.3 million in the 1980 period.