Ashley Smith, debutant Joe Bedford, Steffan Jones, Adam Hughes, Mike Poole and Rhys Buckley all scored for the Dragons as Edwards rotated his squad.
The Bears - preparing for the World Cup with a UK Tour - replied with scores from Kirill Kushnarev and the best try of the evening from Igor Klyuchnikov.
Despite a slow start, the Dragons, who kick off their RaboDirect PRO12 season away at Munster, scored first when centre Smith finished off a spectacular move, but Jason Tovey's conversion hit the post.
This soon put the Dragons in their stride and debutant scrum half Bedford celebrated with a try after excellent work by Gavin Thomas, Tovey converting his first of the night.
Russia scored their first try of the evening when flanker Kushnarev squeezed over to cap off a successful spell of possession for the Bears.
Tovey took advantage of the Dragons having an extra man putting Jones in after Russia lock Alexander Voytov was yellow-carded for preventing release.
After going into the break 19-7 down, the Russians retaliated with the best try of the game when winger Denis Simlikevich evaded a number of tackles and before he put fellow wing Klyuchnikov in to cross in the corner.
The Dragons' made ten changes in one go and this brought a flurry of tries with Hughes, Poole and Buckley all scoring, with fellow replacement Mike Jones converting all three.
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