In the present article, we construct a new family of locally univalent and sense preserving harmonic mappings by considering a suitable transformation of normalized univalent analytic functions defined in the open unit disc D. We present necessary and sufficient conditions for the functions of this new family to be univalent. Apart from studying properties of this new family, results about the convolutions or Hadamard products of functions from this family with some suitable analytic or harmonic mappings are proved by introducing a new technique which can also be used to simplify the proofs of earlier known results on convolutions of harmonic mappings. The technique presented also enables us to generalize existing such results