Water deficiency is one of the most significant abiotic stresses that negatively affects growth and reduces crop yields worldwide. Most research is focused on model plants and/or crops which are most agriculturally important. In this research, drought stress was applied to two drought stress contrasting varieties of Papaver somniferum (the opium poppy), a non-model plant species, during the first week of its germination, which differ in responses to drought stress. After sowing, the poppy seedlings were immediately subjected to drought stress for 7 days. We conducted a large-scale transcriptomic and proteomic analysis for drought stress response. At first, we found that the transcriptomic and proteomic profiles significantly differ. However, the most significant findings are the identification of key genes and proteins with significantly different expressions relating to drought stress, e.g., the heat-shock protein family, dehydration responsive element-binding transcription factors, ubiquitin E3 ligase, and others. In addition, metabolic pathway analysis showed that these genes and proteins were part of several biosynthetic pathways most significantly related to photosynthetic processes, and oxidative stress responses. A future study will focus on a detailed analysis of key genes and the development of selection markers for the determination of drought-resistant varieties and the breeding of new resistant lineages.