The Simulink State-Space block is one of the ways to specify a linear system. The simulation method (or solver) is specified in Simulation, Model Configuration Parameters, Solver. Some of the methods are ode23, ode45, discrete, etc.
Although it is not specified in the document of lsim(), I would assume that lsim() actually uses one of those integration methods or some of the other methods (help ode15s, ode113) to solve the state-space equation and run the simulaiton. Keep in mind that lsim() is for linear system only while other methods could apply to non-linear systems.