The decomposition of fresh crop residues added to soil for agricultural purposes is complex. This is due to different factors that influence the decomposition process. In field conditions, the incorporation of crop residues into soil does not always have a positive effect on aggregate stability. The aim of this study was to investigate the decomposition effects of residues from two different cover crops (Brassica napus var. oleifera and Secale cereale) and one main crop (wheat straw) on soil aggregate stability. A 105-day incubation experiment was conducted in which crop residues were mixed with sandy loam soil at a rate of 6 g C kg−1 of soil. During the incubation, there were five water additions. The decomposition effects of organic matter on soil conditions during incubation were evaluated by determining the soil functional groups; carbon dioxide (CO2), nitrous oxide (N2O), and methane (CH4) emissions; soil microbial biomass carbon (MBC); and water-stable aggregates (WSA). The functional groups of the plant residues and the soil were analyzed using Fourier transform infrared spectroscopy (FTIR) and a double exponential model was used to estimate the decomposition rates. The results show that the decomposition rate of fresh organic materials was correlated with the soil functional groups and the C/N ratio. Oilseed rape and rye, with lower C/N ratios than wheat straw residues, had faster decomposition rates and higher CO2 and N2O emissions than wheat straw. The CO2 and N2O flush at the start of the experiment corresponded to a decrease of soil aggregate stability (from Day 3 to Day 10 for CO2 and from Day 19 to Day 28 for N2O emissions), which was linked to higher decomposition rates of the labile fraction. The lower decomposition rates contributed to higher remaining C (carbon) and higher soil aggregate stability. The results also show that changes in the soil functional groups due to crop residue incorporation did not significantly influence aggregate stability. Soil moisture (SM) negatively influenced the aggregate stability and greenhouse gas emissions (GHG) in all treatments (oilseed rape, rye, wheat straw, and control). Irrespective of the water addition procedure, rye and wheat straw residues had a positive effect on water-stable aggregates more frequently than oilseed rape during the incubation period. The results presented here may contribute to a better understanding of decomposition processes after the incorporation of fresh crop residues from cover crops. A future field study investigating the influence of incorporation rates of different crop residues on soil aggregate stability would be of great interest.