Studies of e+e−→Ds+D(∗)0K− and the P-wave charmed-strange mesons are performed based on an e+e− collision data sample corresponding to an integrated luminosity of 567 pb−1 collected with the BESIII detector at s=4.600 GeV. The processes of e+e−→Ds+D∗0K− and Ds+D0K− are observed for the first time and are found to be dominated by the modes Ds+Ds1(2536)− and Ds+Ds2∗(2573)−, respectively. The Born cross sections are measured to be σB(e+e−→Ds+D∗0K−)=(10.1±2.3±0.8)pb and σB(e+e−→Ds+D0K−)=(19.4±2.3±1.6)pb, and the products of Born cross section and the decay branching fraction are measured to be σB(e+e−→Ds+Ds1(2536)−+c.c.)⋅B(Ds1(2536)−→D∗0K−)=(7.5±1.8±0.7)pb and σB(e+e−→Ds+Ds2∗(2573)−+c.c.)⋅B(Ds2∗(2573)−→D0K−)=(19.7±2.9±2.0)pb. For the Ds1(2536)− and Ds2∗(2573)− mesons, the masses and widths are measured to be M(Ds1(2536)−)=(2537.7±0.5±3.1)MeV/c2,Γ(Ds1(2536)−))=(1.7±1.2±0.6)MeV, and M(Ds2∗(2573)−)=(2570.7±2.0±1.7)MeV/c2,Γ(Ds2∗(2573)−)=(17.2±3.6±1.1)MeV. The spin-parity of the Ds2∗(2573)− meson is determined to be JP=2+. In addition, the process e+e−→Ds+D(∗)0K− are searched for using the data samples taken at four (two) center-of-mass energies between 4.416 (4.527) and 4.575 GeV, and upper limits at the 90% confidence level on the cross sections are determined.