Musk uploaded a brief clip to his Instagram profile on Thursday night showing a chunk of the tunnel.
The camera flies backwards through the tunnel, passes a few workmen, stops at a complicated set of traffic lights, then thrusts forwards again and pans to the logo of The Boring Company, the firm Musk founded to build the tunnels.
In the caption to the post he said the tunnel, a 3.2km prototype under western LA, is now "almost done!". He continued: "Pending final regulatory approvals, we will be offering free rides to the public in a few months."
The footage racked up more than 1 million views in five hours.
Musk also revealed some details about his plans for a full-size Hyperloop transit system connecting major US cities.
In a reply to a fan tweet about the tunnels, Musk said work on a link between New York City and Washington, D.C., had "already started," and that he hoped to start work on a link between Los Angeles and San Francisco in 2019.
Musk's final vision for the tunnel network would see cars, cyclists or pedestrians on the surface hitch a ride on electrified "skates," then be whisked through a network of tunnels to where they need to go. Musk has said that the pods will be able to outpace jet planes.
The LA tunnel is the first to be built. Here is a map of the planned routes in California. The red part is a 10.4km route which is yet to be constructed, while the blue area is a still-more speculative possible route for the final project.