In the last decade, it was shown that it is possible to reconstruct signals with finite rate of innovation (FRI signals) from the samples of their filtered versions. However, when noise is present, the present reconstruction algorithms tend to be low accuracy. In this work, a new sparsity-based reconstruction method for FRI signals is put forward. The streams of Diracs and exponential reproducing kernel are considered. Firstly, the analog time axis is quantified and aligned to grids. Secondly, selecting a finite subset of time delay parameters, the measurement vector is represented as a sparse linear combination of the amplitude parameters. Finally, the sparse solution is calculated by solving an optimization problem under L0 norm. The position of non-zero elements is approximation to the time delays, and the value of non-zero elements is the amplitude. Extensive numerical simulations demonstrate the accuracy and robustness of our method.