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Theoretical models and laboratory experiments show that CH₃OH is efficiently formed on cold grain surfaces through the successive hydrogenation of CO, forming HCO and H₂CO as intermediate species. In cold cores and low UV-field illumination photo-dissociation regions (PDRs) the ices can be released into the gas-phase through nonthermal processes such as photodesorption, which considerably increases their gas-phase abundances. We investigate the dominant formation mechanism of H₂CO and CH₃OH in the Horsehead PDR and its associated dense core. We performed deep integrations of several H₂CO and CH₃OH lines at two positions in the Horsehead, namely the PDR and dense core, with the IRAM-30m telescope. In addition, we observed one H₂CO higher-frequency line with the CSO telescope at both positions. We determined the H₂CO and CH₃OH column densities and abundances from the single-dish observations complemented with IRAM-PdBI high-angular resolution maps (6'') of both species. We compared the observed abundances with PDR models including either pure gas-phase chemistry or both gas-phase and grain surface chemistry. We derived CH₃OH abundances relative to total number of hydrogen atoms of ∼1.2x10^–10 and ∼2.3x10^–10 in the PDR and dense-core positions, respectively. These abundances are similar to the inferred H₂CO abundance in both positions (∼2x10^–10). We find an abundance ratio H₂CO/CH₃OH of ∼2 in the PDR and ∼1 in the dense core. Pure gas-phase models cannot reproduce the observed abundances of either H₂CO or CH₃OH at the PDR position. The two species are therefore formed on the surface of dust grains and are subsequently photodesorbed into the gas-phase at this position. At the dense core, on the other hand, photodesorption of ices is needed to explain the observed abundance of CH₃OH, while a pure gas-phase model can reproduce the observed H₂CO abundance. The high-resolution observations show that CH₃OH is depleted onto grains at the dense core. CH₃OH is thus present in an envelope around this position, while H₂CO is present in both the envelope and the dense core itself. Photodesorption is an efficient mechanism to release complex molecules in low-FUV-illuminated PDRs, where thermal desorption of ice mantles is ineffective.