We investigated methane oxidation in the oxygen minimum zone (OMZ) of the eastern tropical North Pacific (ETNP) off central Mexico. Methane concentrations in the anoxic core of the OMZ reached ~ 20 nmol L−1 at off shelf sites and 34 nmol L−1 at a shelf site. Rates of methane oxidation were determined in ship‐board incubations with 3H‐labeled methane at O2 concentrations 0–75 nmol L−1. In vertical profiles at off‐shelf stations, highest rates were found between the secondary nitrite maximum at ~ 130 m and the methane maximum at 300–400 m in the anoxic core. Methane oxidation was inhibited by addition of 1 μmol L−1 oxygen, which, together with the depth distribution, indicated an anaerobic pathway. A coupling to nitrite reduction was further indicated by the inhibitory effect of the nitric oxide scavenger 2‐phenyl‐4,4,5,5‐tetramethylimidazoline‐1‐oxyl‐3‐oxide (PTIO). Metatranscriptomes from the anoxic OMZ core supported the likely involvement of nitrite‐reducing bacteria of the NC10 clade in anaerobic methane oxidation, but also indicated a potential role for nitrate‐reducing euryarchaeotal methane oxidizers (ANME‐2d). Gammaproteobacteria of the Methanococcales were further detected in both 16S rRNA gene amplicons and metatranscriptomes, but the role of these presumed obligately aerobic methane oxidizers in the anoxic OMZ core is unclear. Given available estimates of water residence time, the measured rates and rate constants (up to ~ 1 yr−1) imply that anaerobic methane oxidation is a substantial methane sink in the ETNP OMZ and hence attenuates the emission of methane from this and possibly other OMZs.