Stochastic model predictive control addresses uncertainties by incorporating the probabilistic description of the disturbances into joint chance constraints. Yet, the classic methods for handling this class of constraints are often computationally inefficient and overly conservative. To overcome this, we propose to replace the nonconvex inverse cumulative distribution function of the standard normal distribution in the deterministic counterpart of these constraints with a highly accurate, exponential cone-representable approximation. This allows the constraints to be formulated as exponential cone functions, and the problem is solved as an exponential cone optimization with risk allocation as decision variables. The main advantage of the proposed approach is that the optimization problem is efficiently solved with off-the-shelf software, and with reduced conservativeness. Moreover, it applies to any problem with linear joint chance constraints subject to normally distributed disturbances. We validate our method with numerical examples of stochastic model predictive control applications.