The reigning three-time world champion flew direct to Sochi, which is also due to hold the 2014 Winter Olympics, following his victory in Sunday's Bahrain Grand Prix.
Although the 3.7-mile track remains under construction, Vettel drove part of the circuit in a road car, the FX Vettel Edition from Red Bull title sponsor Infiniti.
It is the second time in 10 months that Vettel has been the first driver to glimpse a new F1 venue having also drove the planned circuit in New Jersey, which is also due to join the calendar in 2014.
Vettel said: "It is a great privilege to be the first Formula One driver to visit the new venue. After trying a section of the track, I can already see it will be a fun circuit to drive in a Formula One car, and the rest of the venue is very impressive.
"I'm really looking forward to coming back here for the grand prix next year."
The circuit is part of the area's post-Olympic legacy programme, according to Oleg Zabara, deputy general director of the race organisers.
Zabara added: "Construction of the track and the main facilities is on schedule.
"Hosting a Formula One grand prix is a key project for this country, highlighting the status and the image of modern Russia."