It is shown that the $pi^0$ transition form factor $F(Q_1^2, Q_2^2)$ differs substantially from its one-real-photon limit $F(Q_1^2, 0)$ even for rather small values of $Q_2^2 approximate to 0.1 GeV^2$, which cannot be excluded in experiments with one "untagged" electron. It indicates that the comparison of data with theoretical calculations, which usually assume $Q_2^2 = 0$, may be untrustworthy. Our phenomenological model of the $pi^0$ transition form factor is based on the vector-meson-dominance hypothesis and all its parameters are fixed by using the experimental data on the decays of vector mesons. The model soundness is checked in the two-real-photon limit, where it provides a good parameter-free description of the $pi^0 -> 2 gamma$ decay rate, and in the $pi_0$ Dalitz decay. The dependence of $F(Q_1^2, Q_2^2)$ on $Q_1^2$ at several fixed values of $Q_2^2$ is presented and the comparison with existing data performed.