Pulsar wind nebulae (PWNe) studies with the Chandra X-ray Observatory have opened a new window to address the physics of pulsar winds, zoom on their interaction with their hosting supernova remnant (SNR) and interstellar medium, and identify their powering engines. We here present a new 70 ks, plus an archived 18 ks, Chandra ACIS observation of the SNR CTB 87 (G74.9+1.2), classified as a PWN with unusual radio properties and poorly studied in X-rays. We find that the peak of the X-ray emission is clearly offset from the peak of the radio emission by ~100" and located at the southeastern edge of the radio nebula. We detect a point source - the putative pulsar - at the peak of the X-ray emission and study its spectrum separately from the PWN. This new point source, CXOU J201609.2+371110, is surrounded by a compact nebula displaying a torus-like structure and possibly a jet. A more extended diffuse nebula is offset from the radio nebula, extending from the point source to the northwest for ~250" The spectra of the point source, compact nebula and extended diffuse nebula are all well described by a power law model with a photon index of 1.1 (0.7-1.6), 1.2 (0.9-1.4) and 1.7 (1.5-1.8), respectively, for a column density N_H = 1.38 (1.21-1.57) x 10^22 cm^-2 (90% confidence). The total X-ray luminosity of the source is ~1.6 x 10^34 erg s^-1 at an assumed distance of 6.1 kpc, with ~2% and 6% contribution from the point source and compact nebula, respectively. The observed properties suggest that CTB 87 is an evolved (~5-28 kyr) PWN, with the extended radio emission likely a `relic' PWN, as in Vela-X and G327.1-1.1. To date, however, there is no evidence for thermal X-ray emission from this SNR, and the SNR shell is still missing, suggesting expansion into a low-density medium (n_0 < 0.2 cm^-3, assuming D = 6.1 kpc), likely caused by a stellar wind bubble blown by the progenitor star.