> Pauli equation describes the behavior of electron in electromagnetic field and takes into account its spin.
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> Does it explain any experiment, for example Zeeman effect in hydrogen atom? I am looking for a reference where this is explicitly stated and, preferably, quotes the relevant experiment.

Yes. The main evidence that established the existence of electron spin was provided by: (1) the Stern-Gerlach experiment; and (2) the Zeeman effect. See, for example, Messiah "Quantum Mechanics," Vol. 2, Section IV "SPIN" at page 541: "The other is the hypothesis of electron spin. The principal evidence supporting this hypothesis comes from the study of the behavior of complex atoms in a magnetic field (Zeeman effect, Stern-Gerlach experiment)." 

Therefore, if the Pauli equation properly accounts for spin (at least approximately in some limit), it can be said to explain these experiments (at least approximately in some limit).

Furthermore, as [shown explicitly in the link you provided][1], the Pauli equation in a weak magnetic field includes an interaction that looks like:
$$
H_I = \frac{e}{2m}\left(\vec L + 2\vec S\right)\cdot \vec B\;,
$$
where the factor of 2 in front of $\vec S$ is the electron g-factor (which indeed is *approximately* equal to 2).

This kind of interaction Hamiltonian ($H_I$ above) explains the Zeeman effect. See, for example, Bethe H. and Jackiv "Intermediate Quantum Mechanics" at page 169 (Section titled "Zeeman Effect").

For example, when $J=S=1/2$ we find that the $|JM\rangle$ eigenstates (eigenstates of the unperturbed Hamiltonian) are split, at first order in the interaction, by:
$$
\frac{eBM}{m}\;,
$$
where $M=\pm 1/2$.

 


  [1]: https://en.wikipedia.org/wiki/Pauli_equation