Context. Planets form in protoplanetary disks and inherit their chemical compositions. Aims. It is thus crucial to map the distribution and investigate the formation of simple organics, such as formaldehyde and methanol, in protoplanetary disks. Methods. We analyze ALMA observations of the nearby disk-jet system around the T Tauri star DG Tau in the o − H₂CO 3_{1, 2} − 2_{1, 1} and CH₃OH 3_{−2, 2} − 4_{−1, 4} E, 5_{0, 5} − 4_{0, 4} A transitions at an unprecedented resolution of $ {∼}0{{\overset{\prime\prime}{.}}}{15} $, i.e., ∼18 au at a distance of 121 pc. Results. The H₂CO emission originates from a rotating ring extending from ∼40 au with a peak at ∼62 au, i.e., at the edge of the 1.3 mm dust continuum. CH₃OH emission is not detected down to an rms of 3 mJy beam⁻¹ in the 0.162 km s⁻¹ channel. Assuming an ortho-to-para ratio of 1.8−2.8 the ring- and disk-height-averaged H₂CO column density is ∼0.3−4 × 10¹⁴ cm⁻², while that of CH₃OH is < 0.04−0.7 × 10¹⁴ cm⁻². In the inner 40 au no o − H₂CO emission is detected with an upper limit on its beam-averaged column density of ∼0.5−6 × 10¹³ cm⁻². Conclusions. The H₂CO ring in the disk of DG Tau is located beyond the CO iceline (R_CO ∼ 30 au). This suggests that the H₂CO abundance is enhanced in the outer disk due to formation on grain surfaces by the hydrogenation of CO ice. The emission peak at the edge of the mm dust continuum may be due to enhanced desorption of H₂CO in the gas phase caused by increased UV penetration and/or temperature inversion. The CH₃OH/H₂CO abundance ratio is < 1, in agreement with disk chemistry models. The inner edge of the H₂CO ring coincides with the radius where the polarization of the dust continuum changes orientation, hinting at a tight link between the H₂CO chemistry and the dust properties in the outer disk and at the possible presence of substructures in the dust distribution.