We study the field dependence of the entanglement of formation in one- and two-dimensional quantum spin systems displaying a T=0 field-driven quantum phase transition. Making use of exact results and quantum Monte Carlo data for the entanglement of formation, we show how entanglement can be a fundamental tool to understand the quantum critical behavior and to find ``special states'' that usual magnetic observables do not detect. We find that the ground state of anisotropic two-dimensional S=1/2 antiferromagnets in a uniform field takes the classical-like form of a product state for a particular value and orientation of the field, at which the purely quantum correlations due to entanglement disappear. Moreover, the factorized state both in 1D and 2D systems is found to precede the quantum phase transition. It could hence represent a crucial step towards a global rearrangement of the ground state, in view of the quantum phase transition. Finally, we show that the field-induced quantum phase transition present in the models is unambiguously characterized by a cusp minimum in the pairwise-to-global entanglement ratio R, marking the quantum-critical enhancement of "multipartite" entanglement.