Tesla Inc. reported a record quarterly profit as increased output of its Model 3 sedan ended a seven-quarter stretch of red ink for the electric-car maker, defying Wall Street expectations and giving embattled Chief Executive Elon Musk renewed credibility.

The performance, which followed a series of production delays and missteps by Mr. Musk that have roiled Tesla and its stock, delivered on Mr. Musk’s promise that higher Model 3 production would lead to profit during the third quarter. The CEO in a letter to shareholders...