In this paper, we present a coupled experimental/theoretical investigation of pressure effect on the ferromagnetism of LaCrGe3 and LaCrSb3 compounds. The magnetic, electronic, elastic and mechanical properties of LaCrGe3 and LaCrSb3 at ambient condition are studied by first-principles density functional theory calculations. The pressure dependences of the magnetic properties of LaCrGe3 and LaCrSb3 are also investigated. The ferromagnetism in LaCrGe3 is rather fragile with a ferro- to paramagnetic transition at a relatively small pressure (around 7 GPa from our calculations, and 2 GPa in experiments). The key parameter controlling the magnetic properties of LaCrGe3 is found to be the proximity of the Cr DOS to the Fermi surface, a proximity that is strongly correlated to the distance between Cr atoms along the c-axis, suggesting that there would be a simple way to suppress magnetism in systems with one dimensional arrangement of magnetic atoms. By contrast, the ferromagnetism in LaCrSb3 is not fragile. Our calculation results are consistent with our experimental results and demonstrate the feasibility of using first-principles calculations to aid experimental explorations in screening for materials with fragile magnetism.