Performing state-of-the-art ab initio calculations we study the effect of the electric field on magnetic properties of Co and Co-Pt chains on Pt(111). Our studies give clear evidence that an externally applied electric field could permit one to tailor magnetic anisotropy in chains. It is demonstrated that the physics behind this effect is related to the spin-dependent screening of an external electric field at the chains. A strong enhancement of magnetic anisotropy energy in mixed Co-Pt chains, compared to pure Co chains, is found. The interplay between atomic structure and the magnetic anisotropy energy is discussed.