Lagrangian mechanics is useful for deriving the equations of motion for any system in a simple and structured way. Lagrangian mechanics will result in the same equations of motion (which is what describes how everything moves) as Newtonian mechanics, and once you have the equations of motion, you will probably only be using those in your game and not Lagrangian mechanics.
For simple systems in which you know the forces acting on all objects, Newtonian mechanics is probably going to be easier to use. However, for more complex systems, which may involve holonomic constraints, any coordinate system for describing the state of the system (not necesarily a Carteesian), and many parts that may interact with each other in some non-trivial way, the equations of motion may still be obtained relatively easily with Lagrangian mechanics. If your system is unconstrained, all you need to do is to formulate the kinetic energy and potential energy of the system as functions of the generalized coordinates q, as well as their time derivatives. Then you form the Lagrangian as L = K - V, and the Euler–Lagrange equations will be your equations of motion. You may need to work them a little bit to get them in a form that is useful for implementation in a game (you want to isolate the highest time derivative for each generalized coordinate), but that's basically it.
If you have holonomic constraint, you also need to define a function f(q,t) for each constraint for which the constraint is the iso-surface that corresponds to f(q,t)=0. Then you instead form the Lagrangian as L = K - V + a1·f1 + a2·f2 + ... + an·fn, where f1, ..., fn are the functions for your constraints and a1, ..., an are corresponding, unknown coefficients, or so called Lagrange multipliers (basically, the higher these multipliers are, the higher the force from the corresponding constraints are). Then you use the Euler–Lagrange equations as usual, plus equations for your holonomic constraints, which you use to solve for a1, ..., an to make sure that you stay on the isosurface and that the constraints are satisfied (i.e. you find the forces require to make sure you stay where you are supposed to stay). Even if the principle is simple, how to do this in practice can be a bit involved, but one effective solution is described here. These mutipliers will show up in your equations of motion so you just plug in what you solved for them there.