Paper-based microfluidics are an increasingly popular alternative to devices with conventional open channel geometries. The low cost of fabrication and the absence of external instrumentation needed to drive paper microchannels make them especially well suited for medical diagnostics in resource-limited settings. Despite the advantages of paper microfluidics, many assays performed using conventional open channel microfluidics are challenging to translate onto paper, such as bead, emulsion, and cell-based assays. To overcome this challenge, we have developed a hybrid open-channel/paper channel microfluidic device. In this design, wick-driven paper channels control the flow rates within conventional microfluidics. We fabricate these hybrid chips using laser-micromachined polymer sheets and filter paper. In contrast to previous efforts that utilized external, macroscopic paper-based pumps, we integrated micro-scale paper and open channels onto a single chip to control multiple open channels and control complex laminar flow-pattern within individual channels. We demonstrated that flow patterns within the open channels can be quantitatively controlled by modulating the geometry of the paper channels, and that these flow rates agree with Darcy's law. The utility of these hybrid chips, for applications such as bead-, cell-, or emulsion-based assays, was demonstrated by constructing a hybrid chip that hydrodynamically focused micrometer-sized polystyrene beads stably for >10 min, as well as cells, without external instrumentation to drive fluid flow.