Investigation of the resonances X(4140) and X(4274), which were recently confirmed by the LHCb Collaboration , is carried out by treating them as the color triplet and sextet [cs] [(c) over bar (s) over bar] diquark-antidiquark states with the spin-parity J(P) = 1(+), respectively. We calculate the masses and meson-current couplings of these tetraquarks in the context of the QCD two-point sum rule method by taking into account the quark, gluon, and mixed vacuum condensates up to eight dimensions. We also study the vertices X(4140)J/psi phi and X(4274)J/psi phi and evaluate corresponding strong couplings g(X(4140)J/psi phi) and g(X(4274)J/psi phi) by means of the QCD light-cone sum rule method and a technique of the soft-meson approximation. In turn, these couplings contain required information to determine the width of the X(4140) -> J/psi phi and X(4274) -> J/psi phi decay channels. We compare our results for the masses and decay widths of the X(4140) and X(4274) resonances with the LHCb data and alternative theoretical predictions.