LAS VEGAS—Sen. Bernie Sanders won a convincing victory in Nevada’s caucuses on Saturday, a well-timed boost of momentum for his Democratic presidential bid ahead of a potentially decisive series of primaries in 15 states over the next two weeks.

His victory in the third nominating contest—the first in a diverse state—follows what was for him essentially a first-place tie in Iowa’s caucuses and a narrow win in New Hampshire’s primary.

Mr. Sanders’s win is the latest signal the Democratic Party is moving to embrace a greater role for government in the U.S. economy. With entrance polls showing he garnered strong support from Hispanics, it also enhances his argument that he can build a diverse coalition of voters.

“In Nevada, we have just put together a multigeneration, multiracial coalition, which is going to not only win in Nevada, it’s going to sweep this country,” Mr. Sanders said at a rally in San Antonio, Texas, after the results came in.

Entrance polls of caucus participants showed Mr. Sanders won support among a broad range of the electorate, including men, women, Hispanics, non-Hispanic whites, those with and without college degrees, union and nonunion household members. He also carried the most liberal and all but the oldest voters.