In one aspect, a row unit for dispensing agricultural products into a field may include an implement toolbar and a fertilizer-dispensing assembly coupled to the implement toolbar. The fertilizer-dispensing assembly may include a furrow-forming device configured to form a fertilizer-receiving furrow within an inter-row area of the field, with the inter-row area defined between a pair of adjacent post-emergent crop rows of the field in a lateral direction of the implement toolbar. The fertilizer-dispensing assembly may further include a fertilizer applicator configured to dispense a fertilizer into the fertilizer-receiving furrow. Additionally, the row unit may include a seed-dispensing assembly coupled to the implement toolbar. The seed-dispensing assembly may include a furrow-forming device configured to form a seed-receiving furrow within the inter-row area of the field. The seed-dispensing assembly may further include a seed-dispensing device configured to dispense seeds into the seed-receiving furrow.