The sawtooth chain compound CsCo2(MoO4)2(OH) is a complex magnetic system and here, we present a comprehensive series of magnetic and neutron scattering measurements to determine its magnetic phase diagram. The magnetic properties of CsCo2(MoO4)2(OH) exhibit a strong coupling to the crystal lattice and its magnetic ground state can be easily manipulated by applied magnetic fields. There are two unique Co2+ ions, base and vertex, with Jbb and Jbv magnetic exchange. The magnetism is highly anisotropic with the b-axis (chain) along the easy axis and the material orders antiferromagnetically at TN = 5 K. There are two successive metamagnetic transitions, the first at Hc1 = 0.2 kOe into a ferrimagnetic structure, and the other at Hc2 = 20 kOe to a ferromagnetic phase. Heat capacity measurements in various fields support the metamagnetic phase transformations, and the magnetic entropy value is intermediate between S = 3/2 and 1/2 states. The zero field antiferromagnetic phase contains vertex magnetic vectors (Co(1)) aligned parallel to the b-axis, while the base vectors (Co(2)) are canted by 34° and aligned in an opposite direction to the vertex vectors. The spins in parallel adjacent chains align in opposite directions, creating an overall antiferromagnetic structure. At a 3 kOe applied magnetic field, adjacent chains flip by 180° to generate a ferrimagnetic phase. An increase in field gradually induces the Co(1) moment to rotate along the b-axis and align in the same direction with Co(2) generating a ferromagnetic structure. The antiferromagnetic exchange parameters are calculated to be Jbb = 0.028 meV and Jbv = 0.13 meV, while the interchain exchange parameter is considerably weaker at Jch = (0.0047/Nch) meV. Our results demonstrate that the CsCo2(MoO4)2(OH) is a promising candidate to study new physics associated with sawtooth chain magnetism and it encourages further theoretical studies as well as the synthesis of other sawtooth chain structures with different magnetic ions.