This Letter explores the influences of intracluster magnetic fields (≳1 μG) submerged in the hot electron gas on the classic Sunyaev-Zel'dovich effect (SZE) and on the thermal bremsstrahlung in X-ray emissions. As the Larmor frequency is much higher than all collision frequencies, the presence of a magnetic field may lead to an anisotropic velocity distribution of hot electrons. For the two-temperature relativistic Maxwell-Boltzmann distribution, we compute modifications to the classical thermal SZE. Intracluster magnetic fields tend to enhance the SZE with steeper radial variations, which bear important consequences for cluster-based estimates of cosmological parameters. By applying the magnetic SZE theory to spectral observations of SZE and Chandra X-ray emissions from the galaxy cluster A2163, a ~30-40 μG central core magnetic field B0 is predicted. For the SZE and Chandra X-ray spectral observations of the Coma Cluster, our theoretical analysis is also consistent with an observationally inferred B0 ≲ 10 μG. As the magnetic SZE is redshift (z) independent, this mechanism might offer a potentially important and unique way of probing intracluster magnetic fields in the expanding universe.