The most exciting promise of cell micro-bioreactors is their expected ability to reproduce specific aspects of physiological environments in vitro, singularly or in combination, and to perform advanced biological studies to unravel novel interaction mechanisms; but this challenge is formidable. Moreover, the use of microfluidics is limited by the difficulty to adapt standard tissue culture protocols to miniaturised biochips and by the need of bulky external equipment such as liquid pumping systems. Here, we introduce an original microfluidic gradient generator for culturing cells in closed microchambers, based on: (1) automatic cell valving, (2) hydrostatic-pressure pumping, and (3) on-chip liquid reservoirs. The biochip is designed to be fully compatible with standard biological procedures and to operate with no external control units or cumbersome ancillary components. Remarkably, the use of integrated liquid reservoirs prevents air bubble formation and associated channel clogging. We detail the rationale behind the fluidic design, simulate and measure the chip fluid-dynamics by finite element simulations and microflow velocimetry, respectively. We finally exploit our chip to study the kinetics of an anti-cancer molecule, the catechin-dextran conjugate, on HeLa cells, demonstrating a very fast action. We argue that the presented fluidic concept and architecture are not limited to the present realization, but can be easily applied to other chip geometries and adapted to a number of biological studies