The United States Equestrian Team won the $10,000 Nation's Cup as the Washington International Horse Show continued last night at Capital Centre. It marked the 12th time the U.S. team has won the international event here.

The team, which consisted of Olympic gold medalist Joe Fargis, Katie Monahan, Louis Jacobs and Michael Matz, finished with a total of four faults. Scores are based on jumping performances and time penalties incurred by the best three horses in each of two rounds.

In the case of the U.S. team, the final score of Matz on Bon Retour of 23 1/2 faults was dropped from the final calculation. The three remaining riders all completed the 14-jump course without faults. Jacobs had four faults in the first round.

Monahan, 31, of Middleburg, Va., rode The Governor, owned by Mr. and Mrs. Averell Harriman. He is a large brown gelding who began competition on the Grand Prix jumping circuit in the middle of last year. "He's still a little bit green," Monahan said. "But he felt great tonight. Once he gets confident, he's good any place, indoors or out."

Fargis, 37, who grew up in Vienna, Va., and now claims Southhampton, N.Y., as home base, rode his Olympic mount, Touch of Class, a bay mare. Jacobs, 21, of East Aurora, N.Y., rode Janus De Ver. Last night marked the first time Jacobs has represented the team in the United States. He performed with the team last year in Aachen, West Germany.

Second place in the competition, which is conducted only at certain shows, went to the West German team, which finished with 10 1/4 faults. In third place was Canada with 71.5 faults.

In other events yesterday, Patricia Heuckeroth took the first championship offered in the hunter classes. Heuckeroth, of Southern Pines, N.C., rode Holey Sneakers to the championship in the green conformation hunter division after gathering 7 1/2 points.

Heuckeroth also took first place in two other events. In the under-saddle class offered for green conformation hunters, she rode Wave Bonder, a 5-year-old chestnut gelding owned by Mrs. R. Braga and Mrs. J. Carter Brown. In the regular conformation hunter event, Heuckeroth rode Sly Step to the championship.