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As everyone has pointed out, Russian subs have district advantages and disadvantages over American subs. The Yasen had a speed, depth, and maneuverability advantage over the Virginia class. She also have nearly the same number of VLS tubes, and twice the number of torpedo tubes, and also has a more versatile array of weapons (shkval, kalibr and klub lam, asm, Asw missiles, uset 80 and UGST torps). The disadvantage of it compared to a Virginia class is the Virginia is slightly quieter, and has a more capable sonar, and there are more of them.

I for one believe that both these subs are extremely capable platforms, crewed by competent men. As one mentioned, the Russians utilize more automation and have smaller crews. On the one side, this increases the workload on each man if the automation isn't effective, and the possibility of ones mistake being missed is increased. However, this is offset by the fact that smaller crews can be more efficient if automation and training is good, and the amount of supplies and space for the crew can be reduced, meaning smaller submarines, or room for more equipment. Also it means that if your training smaller crews, you can devote more time and money to training them then you do with large ones. You take a class of men. Train them, and see how they do. Only the best will be put in a unit.

All in all, I believe that in a one on one engagement, it would boil down to tactics, and the situation, and the environment. I also believe that that chance of a mutual kill situation where both ships sink each other is a very high possibility.