The incident occurred next to a construction site, near the Windstock restaurant, where cantonal police officers were attempting to check a stolen car and its occupants at around 6am, police said.
It is not clear why the shooting started, although at a press conference prosecutor Paul Schmidig told reporters the Moldovan was about to be arrested for an undisclosed charge.
The car, a black Volkswagen Caravelle, had been reported stolen in another canton the previous day and police had tracked down the vehicle.
It is not clear whether the two men were armed.
Within half an hour of the shooting, at least 100 police officers descended on the scene, including a canine unit, while a helicopter surveyed the area in search of the fugitive, Blick.ch reported.
Schwyz police described the missing man as 170 to 175 centimetres tall, 30 to 35 years old, with short fair hair.
The fugitive was wearing a dark jacket and “is likely to be wet and dirty”, police said.
He was last seen running into the woods.
His colleague “died on the spot,” police said.
The cantonal prosecutor’s office has been called in to investigate the case.
Forensic experts from Zurich and Zug police are also looking into the cause of the death.
According to Schwyz police commander Lorenzo Hutter, members of the force only use firearms as a last resort.
“It is important that the man who fled can be questioned on the unfolding of events,” prosecutor Schmidig said.