The study of cell elasticity provides new insights into not only cell biology but also disease diagnosis based on cell mechanical state variation. Microfluidic technologies have made noticeable progress in studying cell deformation with capabilities of high throughput and automation. This paper reports the development of a novel microfluidic system to precisely measure the elasticity of cells having large deformation in a constriction channel. It integrated i) a separation unit to isolate rod- or flake-shaped particles that might block the constriction channel to increase the measurement throughput and ii) a pressure feedback system precisely detecting the pressure drop inducing the deformation of each cell. The fluid dynamics of the separation unit was modeled to understand the separation mechanism before the experimental determination of separation efficiency. Afterward, the pressure system was characterized to demonstrate its sensitivity and reproducibility in measuring the subtle pressure drop along a constriction channel. Finally, the microfluidic system was employed to study the stiffness of both K562 and endothelial cells. The cell protrusion and pressure drop were employed to calculate the mechanical properties based on a power-law rheology model describing the viscoelastic behaviors of cells. Both the stiffness and the fluidity of K562 and endothelial cells were consistent with those in previous studies. The system has remarkable application potential in the precise evaluation of cell mechanical properties.