By using first-principles calculations based on density-functional theory, the structure stability, magnetic properties and electronic structure of uniformly and nonuniformly alloyed Cu-Co monoatomic chains have been systematically investigated. The effect of elastic stretching/contraction deformations on the stability and electronic properties of alloyed Cu-Co atomic chains has also been studied. The cohesive energy per atom of Cu-Co atomic chain is lower than that for the corresponding ideal Cu atomic chain in the entire average interatomic distance. Thus, the alloyed Cu-Co atomic chain is more stable than the corresponding ideal Cu nanowire. It is established that Co dimers are easily formed in the nonuniformly alloyed Cu-Co atomic chain. The formation of Co2 dimers results in a nonuniformly distribution of the electron density and interatomic distances along the wire direction and leads to accelerated rupture of the wire between Cu atoms under stretching. While the uniformly alloyed Cu-Co atomic chain composed of regularly alternating Cu and Co atoms is stable under stretching up to large interatomic distances. The electronic structure reveale that strong hybridization between the Cu and Co states is responsible for the high stability of alloyed Cu-Co nanowire.