Partial cross sections of the $^{89}$Y(p,$gamma$)$^{90}$Zr reaction have been measured to investigate the $gamma$-ray strength function in the neutron-magic nucleus $^{90}$Zr. For five proton energies between $E_p=3.65$ MeV and $E_p=4.70$ MeV, partial cross sections for the population of seven discrete states in $^{90}$Zr have been determined by means of in-beam $gamma$-ray spectroscopy. Since these $gamma$-ray transitions are dominantly of $E1$ character, the present measurement allows an access to the low-lying dipole strength in $^{90}$Zr. A $gamma$-ray strength function based on the experimental data could be extracted, which is used to describe the total and partial cross sections of this reaction by Hauser-Feshbach calculations successfully. Significant differences with respect to previously measured strength functions from photoabsorption data point towards deviations from the Brink-Axel hypothesis relating the photo-excitation and de-excitation strength functions.