The effective spin-1/2 antiferromagnetic Heisenberg-Ising chain materials, ACo₂V₂O8, A = Sr, Ba, are a rich source of exotic fundamental phenomena and have been investigated for their model magnetic properties both in zero and nonzero magnetic fields. Here we investigate a new member of the family, namely, PbCo₂V₂O₈. We synthesize powder and single-crystal samples of PbCo₂V₂O₈ and determine its magnetic structure using neutron diffraction. Furthermore, the magnetic field/temperature phase diagrams for a magnetic field applied along the c, a, and [110] crystallographic directions in the tetragonal unit cell are determined via magnetization and heat capacity measurements. A complex series of phases and quantum phase transitions are discovered that strongly depend on both the magnitude and direction of the field. Our results show that PbCo₂V2O₈ is an effective spin-1/2 antiferromagnetic Heisenberg-Ising chain with properties that are, in general, comparable to those of SrCo₂V2O₈ and BaCo₂V₂O₈. One interesting departure from the results of these related compounds is, however, the discovery of a new field-induced phase for the field direction H ∥[110].