All it took to draw the U.S. Justice Department into investigating Tesla Inc. was a single tweet by Chairman Elon Musk. But now that prosecutors have a toehold, they can dig in to look for other signs of misconduct at the electric car maker.

Tesla Inc. is under investigation by the Justice Department over public statements made by the company and CEO Elon Musk, according to two people familiar with the matter. The criminal probe is running alongside a previously reported civil inquiry by securities regulators.