The estimation of the directions-of-arrival (DOAs) of multiple sound sources is a fundamental stage in acoustic scene analysis. Many application areas such as robot audition and object-based audio (OBA) broadcast require that DOA estimation is computationally efficient to allow real-time operation. We propose a new DOA estimation approach based on a sparse representation of recorded sound fields as a linear combination of spatially bandlimited impulses in this paper. The proposed algorithm operates on a time-frequency representation of the spherical harmonic components of the sound field. We describe a residual energy test that can identify time-frequency bins with a single active source. DOA estimation is carried out at each time-frequency bin by seeking a single-source dictionary atom which provides the best match to the steered response function calculated at the selected bins. We demonstrate the accuracy of the proposed method via a set of emulations using acoustic impulse responses measured in a highly reverberant room.