Using a total of 9.0 fb^{-1} of e^{+}e^{-} collision data with center-of-mass energies between 4.15 and 4.30 GeV collected by the BESIII detector, we search for the processes e^{+}e^{-}→γX(3872) with X(3872)→π^{0}χ_{cJ} for J=0, 1, 2. We report the first observation of X(3872)→π^{0}χ_{c1}, a new decay mode of the X(3872), with a statistical significance of more than 5σ for all systematic fit variations. Normalizing to the previously established process e^{+}e^{-}→γX(3872) with X(3872)→π^{+}π^{-}J/ψ, we find B(X(3872)→π^{0}χ_{c1})/B(X(3872)→π^{+}π^{-}J/ψ)=0.88_{-0.27}^{+0.33}±0.10, where the first error is statistical and the second is systematic. We set 90% confidence level upper limits on the corresponding ratios for the decays to π^{0}χ_{c0} and π^{0}χ_{c2} of 19 and 1.1, respectively.