Southbound traffic will then start using the tunnel, with northbound traffic due to start using it in a few days.
Construction began in January 2007.
The twin-bore tunnel, one of the longest in England, is designed to relieve the Hindhead traffic bottleneck.
Viewing platforms were built during its construction to allow spectators to monitor progress.
The opening ceremony will not be open to the public for "safety and operational reasons", the Highways Agency said.
The tunnel, which will be used by 30,000 vehicles a day, runs under the bowl, which is a large hollow of dry, sandy heath, to the east of Hindhead.
The existing A3, between the National Trust cafe and Boundless Road, will be closed to through-traffic after the tunnel is opened in both directions.