A novel approach to proton CT reconstruction using backprojection-then-filtering (BPF) is proposed. A list-mode algorithm is formulated accommodating non-linear proton paths. The analytical form is derived for the deblurring kernel necessary for the filtering step. Further, a finite matrix correction is derived to correct for the limited size of the backprojection matrix. High quantitative accuracy in relative stopping power is demonstrated (⩽0.1%) using Monte Carlo simulations. This accuracy makes the algorithm a promising candidate for future proton CT systems in proton therapy applications. For the purposes of reconstruction, each proton path in the object-of-interest was estimated based on a cubic spline fit to the proton entry and exit vectors. The superior spatial-resolution of the BPF method over the standard filtering-then-backprojection approach is demonstrated. As the BPF algorithm requires only one backprojection and filtering operation on a scan data set, it also offers computational advantages over an iterative reconstruction approach.