We present a new calculation of the semileptonic tree-level and flavor-changing neutral current form factors describing B-meson transitions to tensor mesons T = D-2*, K-2*, a(2), f(2) (J(P) = 2(+)). We employ the QCD light-cone sum rules approach with B-meson distribution amplitudes. We go beyond the leading-twist accuracy and provide analytically, for the first time, higher-twist corrections for the two-particle contributions up to twist-four terms. We observe that the impact of higher-twist terms to the sum rules is noticeable. We study the phenomenological implications of our results on the radiative B -> K-2(*)gamma and semileptonic B -> D-2*l (v) over bar (l), B -> K-2*l(+)l(-) decays.