The Varshavyanka class, with more advanced stealth technology and an extended combat range, is an improvement on its predecessor, the Kilo class of submarines, RIA Novosti said.
The U.S. Navy calls the diesel-electric subs black-holes because their technology makes them nearly undetectable when submerged.
Vietnam will use the 3,100-ton submarines primarily for anti-shipping and anti-submarine missions in relatively shallow waters.
RIA Novosti said Vietnam ordered the six submarines in 2009 to counter China's increasing naval influence in the region. The contract, worth around $2 billion, includes training of Vietnamese crews in Russia.
A Vietnamese crew has been training since April in the first of the submarines, the Novorossiisk, that completed its 100-day sea trials last month and is expected to be delivered in November, Admiralty Shipyards said.
Admiralty will float out Vietnam's fourth Varshavyanka-class vessel in November, with delivery of the last vessel to Vietnam scheduled for 2016.
A report by Naval-Technology in August last year said the Varshavyanka-class vessels have a range of 400 miles using electronic propulsion and can patrol for 45 days.
The submarines carry a crew of 52 and feature 533mm torpedo tubes and Kalibr 3M54 (NATO SS-N-27 Sizzler) cruise missiles, developed by Russia's Novator Design Bureau.
Top speed is 20 knots and diving depth is about 985 feet.
The shipyard also said the first of another six being built for Russia's Black Sea Fleet would be floated out in November.
Construction of the first Varshavyanka-class sub for the Black Sea Fleet, a vessel named the Novorossiisk, began in 2010 and was completed ahead of schedule, the shipbuilder said.
Admiralty Shipyard also constructed India's ill-fated Sindhurakshak submarine that was destroyed by an explosion this month.
The vessel is the seventh and last of the 10 Kilo-class diesel-electric submarines the Indian navy acquired from Moscow between 1986 and 2000.
Admiralty built the Sindhurakshak in 1997 and the Russian Severodvinsk Zvezdochka shipyard in Severodvinsk recently completed a 2 1/2-year, $80 million modernization of the vessel.
Sindhurakshakat had arrived at Mumbai Naval Docks on April 29 but sank after a massive explosion Aug. 14 that killed 18 seamen.