In this paper, we study the twisted Fourier-Mukai partners of abelian surfaces. Following the work of Huybrechts [19], we introduce the twisted derived equivalence between abelian surfaces. We show that there is a twisted derived Torelli theorem for abelian surfaces over fields with characteristic = 2. Over complex numbers, twisted derived equivalence corresponds to rational Hodge isometries between the second cohomology groups, which is in analogy to the work of Huybrechts and Fu-Vial on K3 surfaces. Their proof relies on the global Torelli theorem over C, which is missing in positive characteristics. To overcome this issue, we extend Shioda's trick [39] on singular cohomology groups to étale and crystalline cohomology groups and make use of Tate's isogeny theorem to give a characterization of twisted derived equivalence on abelian surfaces via using so called principal quasi-isogeny.