Mr Wetherbee parked the shuttle in front of Mir at 1920 GMT, and the two craft travelled in tandem at 17,500 mph (28,000 kph) from darkness into sunlight high above the southern tip of Kamchatka and the northern tip of Japan.
They maintained the position for about 10 minutes, then the shuttle backed away from Mir and circled the space station at a distance of 400ft (120m) while crew members made a photographic survey.
"This is almost like a trip home for Vladimir Titov," Kari Fluegel, the flight commentator, said at Nasa's Mission Control in Houston, Texas. The manoeuvre was to last about two hours. The crews sighted each other with television cameras about 1812 GMT. The Mir cosmonauts beamed down live pictures of Discovery flying beside the space station.
Moments later, the Discovery crew beamed down images of Mir, a T-shaped cluster of four modules.
Less than three hours before the delicate dance was to begin, the Russian Space Agency granted its approval for Discovery to approach closely.
The decision came after three days of negotiations between US and Russian engineers. The Russians had feared the nine-year-old space station would be damaged by fuel leaking from a steering jet on the shuttle.
US engineers agreed to have the astronauts stop the leak by closing a manifold that supplies fuel to the broken jet.
The crew's excitement was evident yesterday as Mr Titov scanned the heavens for a glimpse of the space station that was his home for a year and a day in 1987 and 1988.
"That old eagle-eyes thinks he may have seen Mir," Mr Wetherbee radioed Mission Control five hours before Discovery began its final approach.
Mr Titov, assigned to handle communications between the shuttle and his Russian-speaking colleagues, made radio contact when the two craft were 104 miles (167 km) apart.
The rendezvous is intended as a pathfinder for future shuttles, the first in June, that will dock at Mir to transfer equipment and crew.