Many fonts on Typekit support Russian. You can filter for fonts that support Russian by selecting "Russian" from the Language Support section on http://typekit.com/fonts.
After you have added a font to your kit, choose the "All Characters" option or create a custom language subset which includes Russian to add the language to your kit. Republish your kit, and the Russian characters will be available to your website within a few minutes.
The Language Support & Subsetting help page has more information on the character set options.
All fonts that support Russian include the following letter characters, at minimum: