Mapping observations of the J=6-->5 transition of N2O and the Π+1/2, J=3/2-->1/2 line of NO in the 2 mm band toward the core region of the Sagittarius B2 complex have been carried out using the Kitt Peak 12 m telescope. Emission from NO was found to be extended over a region 2'×5' in size that includes the Sgr B2 (N), Sgr B2 (M), and Sgr B2 (OH) positions, very similar to the distribution found for HNO. In contrast, N2O emission was confined to a source approximately 1' in extent, slightly elongated in the north-south direction and centered on the Sgr B2 (N) core. A virtually identical distribution was found for the JKτ=140-->14-1 E transition of methanol, which lies 255 K above ground state and samples very hot gas. Excitation conditions are favorable for the J=6-->5 line of N2O over the entire NO region; hence, the confined nature of this species is a result of chemistry. The J=3-->2 and J=9-->8 lines of N2O at 75 and 226 GHz, respectively, were also detected at Sgr B2 (N). Combined with the J=6-->5 data, these transitions indicate a column density for this molecule of Ntot~1.5×1015 cm-2 at this position and an abundance of f(N2O/H2)~1.5×10-9. This fractional abundance is almost 2 orders of magnitude higher than predicted by low-temperature chemical models. The N2O observations suggest that this molecule is preferentially formed in high-temperature gas; a likely mechanism is the neutral-neutral reaction NO+NH-->N2O+H, which has an appreciable rate only at T>125 K. The column density of NO found over the Sgr B2 cloud was Ntot~(0.8-1.5)×1016 cm-2, corresponding to a fractional abundance of f(NO/H2)~(0.8-1.5)×10-8, which is about 1 order of magnitude less than model predictions. The similar distributions of NO and HNO suggest a chemical connection. It is likely that the major route to HNO is from NO via the ion-molecule process NO+HNO+-->NO++HNO, which occurs readily at low temperatures. The NO molecule thus appears to be the main precursor species in the N/O chemical network.