Authorities did not mention terrorism as a possible cause.
According to eyewitnesses, the bus swerved into a pedestrian area, where people were walking.
Footage shows people scattering as the bus ran down wide steps before being brought to a halt by the tunnel's roof.
It is not clear what caused the crash but Russian media said the driver had been detained after he lost control of the vehicle.
Traffic spokesperson Artyom Kolesnikov confirmed: "A bus drove into a pedestrian underpass. Police employees are working with him", the statement said.
The accident of a kind not heard of in Moscow in decades occurred near the Slavyansky Boulevard metro station at around 14:50.
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He tried to apply the brakes but they did not work, the investigative committee said.
Officials said there is no suggestion it was a terror attack while Russian investigators said the driver told police the vehicle had started to move suddenly.
Sirens were blaring around the metro station, with authorities preparing to pull the bus out of the underpass with a cable, an AFP correspondent at the scene said.
The emergencies ministry said two helicopters had been dispatched to the scene to help evacuate the victims.
Sobyanin has vowed to conduct a "thorough" probe into the deadly accident.