(abridged) The first unidentified very high energy gamma ray source (TeV J2032+4130) in the Cygnus region has been the subject of intensive search for a counterpart source at other wavelengths. A deep ($approx 50$ ksec) exposure of TeV J2032+4130 with textit{XMM-Newton} has been obtained. The contribution of point sources to the observed X-ray emission from TeV J2032+4130 is subtracted from the data. The point-source subtracted X-ray data are analyzed using blank sky exposures and regions adjacent to the position of TeV J2032+4130 in the field of view covered by the XMM-Newton telescopes to search for diffuse X-ray emission. An extended X-ray emission region with a full width half maximum (FWHM) size of $approx 12$ arc min is found. The centroid of the emission is co-located with the position of TeV J2032+4130.The energy spectrum of the emission coinciding with the position and extension of TeV J2032+4130 can be modeled by a power-law model with a photon index $Gamma=1.5pm0.2_mathrm{stat}pm0.3_mathrm{sys}$ and an energy flux integrated between 2 and 10 keV of $f_{2-10 mathrm{keV}} approx 7cdot 10^{-13}$ ergs/(cm$^2$ s) which is lower than the very high energy gamma-ray flux observed from TeV J2032+4130. We conclude that the faint extended X-ray emission discovered in this observation is the X-ray counterpart of TeV J2032+4130. Formally, it can not be excluded that the extended emission is due to an unrelated population of faint, hot ($k_BTapprox 10$ keV) unresolved point-sources which by chance coincides with the position and extension of TeV J2032+4130. We discuss our findings in the frame of both hadronic and leptonic gamma-ray production scenarios.
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