A higher-order numerical technique is presented for the direct sensitivity analysis of the shallow water equations with passive scalar transport. The continuous sensitivity equations are modified to account for the possible presence of shocks in the solution, that result in Dirac source terms for the sensitivity across flow discontinuities. Higher-order accuracy is achieved via a MUSCL reconstruction technique with slope limiting, which makes the numerical solution Total Variation Diminishing (TVD). The Harten-Lax-van Leer (HLL) approximate Riemann solver is modified so as to account for the influence of source terms in both the flow and sensitivity solutions. Several options are tested for the wave speed estimates and the order of the MUSCL time stepping, such as the MUSCL- Hancock, MUSCL-EVR and MUSCL-HLLG techniques. Convergence analyses on continuous and discontinuous flow problems with analytical solutions indicate that first-order time stepping is approximately twice as fast as second-order time stepping and that it yields more accurate sensitivity solutions.