Using a data sample collected with the BESIII detector operating at the BEPCII storage ring at a center-of-mass energy of $\sqrt{s}$=4.600 GeV, we search for the production of $e^{+}e^{-} \to \phi\chi_{c0,1,2}$ and the charmonium-like state $Y$(4140) in the radiative transition $e^{+}e^{-} \to \gamma Y(4140)$ with $Y$(4140) subsequently decaying into $\phi J/\psi$. The processes $e^{+}e^{-} \to \phi\chi_{c1}$ and $\phi\chi_{c2}$ are observed for the first time, each with a statistical significance of more than 10\sigma, and the Born cross sections are measured to be $(4.2^{+1.7}_{-1.0}\pm 0.3)$ pb and $(6.7^{+3.4}_{-1.7}\pm 0.5)$pb, respectively, where the first uncertainties are statistical and the second systematic. No significant signals are observed for $e^{+}e^{-} \to \phi\chi_{c0}$ and $e^{+}e^{-} \to \gamma Y(4140)$ and upper limits on the Born cross sections at 90% confidence level are provided at $\sqrt{s}$=4.600 GeV.