Another officer in a conventional car took over the pursuit.
Police said the Tesla's battery went flat because an officer on a previous shift forgot to plug it in.
California's Mercury News, which first reported the story, said the Tesla is part of a trial to see if electric cars can replace petrol and diesel-powered vehicles on government fleets.
The publication quoted Fremont Police captain Sean Washington as saying the car performs impressively, and that "we are easily able to make it through an 11-hour shift with battery power to spare".
Victoria Police became the first Australian jurisdiction to field an electric highway patrol car by putting a Tesla Model X to work in June.
Highway patrol officer Inspector Stuart Bailey told the Herald Sun that the Tesla is "the fastest car in our fleet".
"I can see in the future our entire fleet is electric,'' Inspector Bailey said.
"It's coming and it's coming rapidly.
"We're putting ourselves in a great position so when it does come we've done all the research and development and we're ready for the changeover without affecting our capability."