This paper is focused on robust \(H_\infty\) control of a linear time-varying uncertain stochastic system of the form \[ \dot x(t) =(A+\Delta A(t))x(t) +(B+\Delta B(t))u(t) +Dw(t),\quad x(t_0)=x_0, y(t)=Cx(t), \] where \(x(t)\) is the state vector, \(u(t)\) is the control input vector, \(y(t)\) is the measured output vector, and \(A\), \(B\), \(C\), and \(D\) are known constant matrices; \(w(t)\) is a zero-mean Gaussian white noise with covariance \(I>0\); \(x_0\) is the unknown random zero-mean initial state uncorrelated with \(w(t)\); \(\Delta A(t)\) and \(\Delta B(t)\) are time-varying matrices representing parametric perturbations in the system and control matrices. The purpose of the paper is to design a state feedback controller such that, for all admissible time-varying perturbations, the steady-state variance of each state and the \(H_\infty\) norm of the transfer function from disturbance \(w(t)\) to system output \(y(t)\) do not exceed the prespecified upper bounds. Conditions for the existence of expected robust controllers are derived in terms of positive definite solutions to certain parameter-dependent algebraic Riccati equations. Two numerical examples are given to show the effectiveness of the approach used.