A new sensitivity analysis scheme is presented based on explicit expressions for sensitivity coefficients to estimate timewise varying heat flux in heat conduction problems over irregular geometries using the transient readings of a single sensor. There is no prior information available on the functional form of the unknown heat flux; hence, the inverse problem is regarded as a function estimation problem and sensitivity and adjoint problems are involved in the solution of the inverse problem to recover the unknown heat flux. However, using the proposed sensitivity analysis scheme, one can compute all sensitivity coefficients explicitly in only one direct problem solution at each iteration without the need for solving the sensitivity and adjoint problems. In other words, the functional form of the unknown heat flux can be numerically estimated by using the parameter estimation approach. In this method, the irregular shape of heat-conducting body is meshed using the boundary-fitted grid generation (elliptic) method. Explicit expressions are given to compute the sensitivity coefficients efficiently and the steepest-descent method is used as the minimization method to minimize the objective function and reach the solution. Three test cases are presented to confirm the accuracy and efficiency of the proposed inverse analysis.