An unexpectedly high black turnout Tuesday carried Puerto Rican-born Mayor Maurice Ferre to a surprisingly easy fifth-term victory that frustrated the Cuban community's hope of gaining political control of Miami by electing a mayor and a majority of the city commission.

Ferre won 56 percent of the vote in the runoff election to trounce a strong challenger, Manolo Reboso, the city's first elected Cuban-born commissioner who had hoped to become the nation's first Cuban-born mayor.

The city-wide turnout of 51 percent was a record for a municipal election here because blacks turned out to elect a black commissioner. In an election that turned on ethnic politics, Ferre won in his traditional way--with overwhelming black support, a solid third of the Latin vote and roughly half the non-Latin whites.

Reboso, a Bay of Pigs veteran and close friend of the late Nicaraguan dictator Anastasio Somoza, conceded that overkill in Little Havana hurt him. He said that instead of flying to Washington last Thursday to have his photo taken with President Reagan, "maybe it was a mistake of mine not going to the black community."

Ferre credited the black vote, of which he claimed 92 percent, for his victory. "You name me another white man in the history of this town that got that kind of support from the black community," the mayor said.

Spurred by the threatened loss of the so-called "black seat" on the five-member city commission and fear that Reboso was unconcerned about the black community, blacks turned out a record 51 percent.

They elected Miller Dawkins, a black community college administrator, over Cuban American Luis Morse to retain the seat that has been held by a black for 15 years.

In the feverish final week of the campaign, two black radio stations engaged in nearly round-the-clock get-out-the-vote drives. On radio stations Monday, Atlanta's new black mayor, Andrew Young, urged support for Ferre, and disc jockeys created an emotional groundswell by playing taped speeches of the late Martin Luther King.

At stake was "political survival," said Marvin Dunn, a psychologist and talk show host.

Bolstered by their new status as the biggest ethnic group on the voter registration rolls, Cubans originally appeared headed for a sweep of all three city commission seats up for election. In the third race, Demetrio Perez defeated another Cuban American so narrowly that the apparent loser, lawyer Xavier Suarez, asked for a recount.

Most pollsters had written off the impact of the Anglo vote because there was no Anglo in the runoff. Yet, Anglo turnout was 38 percent, higher than in last week's general election. Interviews with voters indicated that Anglos wanted to help preserve the biracial tri-ethnic balance on the commission.

The election came one year after Anglo voters, in an angry backlash against growing Latin influence, overwhelmingly banned the Dade County government's official bilingual policy.

Ferre and Reboso traded their bitterest ethnic charges over Spanish radio stations. Reboso made an unabashed pitch to "vote Cuban" and accused Ferre of being soft on Fidel Castro. Ferre softly pointed out that Reboso was only "in Miami" during the Bay of Pigs invasion. Actually Reboso was on an invasion-force ship that failed to arrive in time.

Meanwhile, Ferre, who speaks flawless English, criticized Reboso, who does not, for failing to show up for several scheduled debates in English during the last days of the campaign.

That made it appear that Reboso was addressing his campaign exclusively to the Latin community.