In 1911, Chavez married Imelda Espinosa, a member of a prominent New Mexico family. In 1914, they moved to Belen. He worked briefly as editor of a Belen weekly newspaper, as a court interpreter, and as a private contractor until 1916, when he obtained temporary employment as a Spanish interpreter for Senator Andrieus A. Jones' election campaign. In 1917, he was offered a position as assistant executive clerk of the Senate in Washington, D.C. by Senator Jones. He accepted this position, passed a special admission exam at Georgetown University Law Center and studied law at night. He graduated from Georgetown in 1920 and returned to Albuquerque to establish a law practice.
He was the first native-born Hispanic elected to the U. S. Senate, and only the second Hispanic in its history. Chavez was a minority politician in the Senate, at a time when his ethnicity was regarded with contempt and racism by many. He was also the first person born in New Mexico elected by the state to the Senate. By a large margin, he is the longest serving Hispanic U.S. Senator.
He accomplished many notable things, but the way New Mexicans remember him is perhaps the most impressive of all. People would travel dozens of miles in the state just to hear him speak and he always amazed them by remembering his constituents' names years after he met them. He was an endearing and warm figure, who was easily likable and seemed very driven. This is perhaps why he is regarded so highly by those who can remember him.
A granddaughter, Gloria Tristani, followed in his footsteps in public service, serving as chair of the New Mexico State Corporations Commission in 1996, a member of the Federal Communications Commission from 1997 to 2001 and as the Democratic candidate to the United States Senate seat from New Mexico in the 2002 elections.