Strategic and Tactical Submarine warfare is highly developed from decades of development training and practice.
The US is quite adept and holds a lead in most technology and the number of boats, with the Russians following closely in technology and farther in numbers.
To oversimplify a bit, the tactical measure is to be as sneaky as possible, and the countermeasure is massive intelligence and observational technology for tracking.
In short, we try to know where every Russian boat is at all times, and they try to know where every American boat is at all times.
When someone pulls the trigger, the Captains act as ordered.
If an American boat were to launch, and it was within range of a shadowing Russian Asset, the Commander of the Russian Asset would perform as ordered. If the American boat was sufficiently out of range, the Russian asset would pursue if ordered, or target other American Assets, if ordered.
The rest depends on skill, quality and quantity, with the US having an advantage in training and quantity.
After eight years of Obama perfumed princes preaching social justice rather than actual ops, can we claim that as an advantage?