Now this is a sticky question because it depends on who you ask and oftentimes what agenda that person is trying to push. Just how far back the history of present day Russia goes has always been a topic of debate, especially where you consider the "cradle" of Russian society to be. In the interest of full disclosure I will say that I usually start tracing the history of Russia from the Grand Duchy of Muscovy, not Kiev. I do this because the Kievan origin is also the origin of Ukraine and Belorus whereas Muscovy is, at least to my mind, the start of the Russian cultural ethnicity.

The Kievan Rus is an interesting topic because the origins of Kiev are somewhat uncertain (if you want more on that, I wrote something about it a while back). This also touches on how Russian the Kievan Rus were and again, it comes down to who you ask and what view they subscribe to.

Personally I subscribe to the non-anti-Normanist view proposed by F. Donald Logan in The Vikings in History basically stating that "...in 839, the Rus' were Swedes. In 1043, the Rus' were Slavs.". Other than that the non-anti-Normanist view basically says that the Kievan Rus were not ethnically anything. They were a mix of people who traveled, traded, raided and lived along the Dnieper river.

However, all of this is really a sidenote because both the Kievan Rus and Muscovy became subjects to the Mongols. The difference is that the Kievan Rus society was sundered by the conquest whereas Muscovy became a collaborator state. It was with Mongol help that Muscovy crushed the rival state of Tver and because the Mongols generally left Muscovy alone it was able to attract people and trade, gradually builing its strength.

So were the Russians conquered by the mongols? Yes. Whether you want to place the start in Muscovy or Kiev they were both Mongol subjects.