We investigate the effect of Co²⁺ (spin-1/2) impurities on the magnetic ground state and low-lying spin excitations of the quasione-dimensional spin-1/2 antiferromagnet SrCuO₂ by means of neutron scattering, muon spin spectroscopy, and bulk (ac and dc) magnetic susceptibilities. We found that dilute Co doping induces an Ising-like anisotropy and enhances the magnetic ordering temperature rather significantly, but preserves the gapless nature of the spin excitations. These results are in apparent contradiction with the recent studies of Ni (spin-1) doped SrCuO₂. Low-temperature magnetic behavior of the Co-doped zigzag chains in SrCuO₂ reveals the presence of a weak geometrical spin frustration.