We obtained X-ray XMM-Newton observations of the open cluster Berkeley 87 and the massive star-forming region (SFR) ON 2. In addition, archival infrared Spitzer Space Telescope observations were used to study the morphology of ON 2, to uncover young stellar objects, and to investigate their relationship with the X-ray sources. It is likely that the SFR ON 2 and Berkeley 87 are at the same distance, 1.23 kpc, and hence are associated. The XMM-Newton observations detected X-rays from massive stars in Berkeley 87 as well as diffuse emission from the SFR ON 2. The two patches of diffuse X-ray emission are encompassed in the shell-like H II region GAL 75.84+0.40 in the northern part of ON 2 and in the ON 2S region in the southern part of ON 2. The diffuse emission from GAL 75.84+0.40 suffers an absorption column equivalent to AV ≈ 28 mag. Its spectrum can be fitted either with a thermal plasma model at T ≳ 30 MK or by an absorbed power-law model with γ ≈ −2.6. The X-ray luminosity of GAL 75.84+0.40 is LX ≈ 6 × 1031 erg s−1. The diffuse emission from ON 2S is adjacent to the ultra-compact H II (UCH II) region Cygnus 2N, but does not coincide with it or with any other known UCH II region. It has a luminosity of LX ≈ 4 × 1031 erg s−1. The spectrum can be fitted with an absorbed power-law model with γ ≈ −1.4. We adopt the view of Turner & Forbes that the SFR ON 2 is physically associated with the massive star cluster Berkeley 87 hosting the WO-type star WR 142. We discuss different explanations for the apparently diffuse X-ray emission in these SFRs. These include synchrotron radiation, invoked by the co-existence of strongly shocked stellar winds and turbulent magnetic fields in the star-forming complex, cluster wind emission, or an unresolved population of discrete sources.