We present a comprehensive mechanism for the emergence of strange attractors in a two-parametric family of differential equations acting on a three-dimensional sphere. When both parameters are zero, its flow exhibits an attracting heteroclinic network (Bykov network) made by two 1-dimensional and one 2-dimensional separatrices between two hyperbolic saddles-foci with different Morse indices. After slightly increasing both parameters, while keeping the one-dimensional connections unaltered, we focus our attention in the case where the two-dimensional invariant manifolds of the equilibria do not intersect. Under some conditions on the parameters and on the eigenvalues of the linearization of the vector field at the saddle-foci, we prove the existence of many complicated dynamical objects, ranging from an attracting quasi-periodic torus, Newhouse sinks to Hénon-like strange attractors, as a consequence of the Torus Bifurcation Theory (developed by Afraimovich and Shilnikov). Under generic and checkable hypothesis, we conclude that any analytic unfolding of a Hopf-zero singularity (within an appropriate class) contains strange attractors.