Bolivia’s electoral tribunal announced late Friday that President Evo Morales had won this week’s presidential election by a margin large enough to avoid a runoff, a finding that incensed the opposition and raised international doubts about a race that has been dogged by allegations of fraud.

At the end of a tumultuous week during which thousands of Bolivians protested what they saw as a power grab by Mr. Morales, election officials said the president had received 47 percent of Sunday’s vote, beating his main rival, Carlos Mesa, by slightly more than 10 percentage points. That was the margin the incumbent needed to avoid a runoff.

The president of the electoral tribunal acknowledged that the results would be received with skepticism by many at home and abroad.

“We are aware of the critical situation we find ourselves in,” said the tribunal’s president, María Eugenia Choque. She added that election officials were willing to have their work audited. “With my heart in my hand,” she said, “we are open to any audit.”