Between 2015 and 2018, we collected approximately 2,000 water column measurements of methane (CH4) and nitrous oxide (N2O) concentrations in the North American Arctic Ocean during summer and early fall. We also obtained 25 measurements of CH4 and N2O concentrations in rivers along the Northwest Passage and Ellesmere Island in midsummer 2017–2019. Our results show that N2O is generated in the highly productive Bering and Chukchi Seas and transported northeastward, producing a persistent subsurface N2O peak in the Beaufort Sea. The Chukchi and Beaufort Sea sediments are a significant source of CH4 to the water column. These sedimentary sources and associated water column consumption display significant spatial gradients and interannual variability. CH4 isotope data demonstrate the importance of CH4 oxidation across the study region. We find that rivers are not a significant source of CH4 or N2O to the Arctic Ocean at the time of year sampled. The estimated annual sea‐air flux across the study region (2.3 million km2) had a median (first quartile, third quartile) of 0.009 (0.002, 0.023) Tg CH4 y−1 and −0.003 (−0.013, 0.010) Tg N y−1. These results suggest that the North American Arctic Ocean currently plays a negligible role in global CH4 and N2O budgets. Our expansive data set, with observations at many repeat stations, provides a synopsis of present‐day Arctic CH4 and N2O distributions and their range of variability, as well as a benchmark against which future climate‐dependent changes can be evaluated.