The interaction between a jet and a stratified cloud (of characteristic size much larger than the jet radius) leads to a final configuration in which the jet has bored a hole through the cloud. This interaction results in a change of the direction, velocity, density and diameter of the jet beam. A simple model, based on Bernoulli's theorem, can be integrated analytically for the case of a plane-parallel, exponential cloud pressure stratification. This model shows that a substantial deflection of the jet beam can be obtained for the adiabatic case (relevant for extragalactic jets), with the jet eventually emerging upwards from the stratified cloud with characteristics (e.g. velocity, density, temperature and diameter) which are basically identical to the ones of the incident jet beam. However, for the radiative case [relevant for Herbig-Haro (HH) jets], a smaller deflection is obtained, with the jet beam eventually becoming almost parallel to the isobars of the plane-parallel cloud stratification. We also find that while a low Mach number jet (with M_0~1) changes direction over a distance of a few environmental pressure scaleheights, a high Mach number (M_0~10) jet is deflected only over distances of many pressure scaleheights. Because of this, high Mach number jets will go through stratified clouds with depths of only a few pressure scaleheights without an appreciable change of direction. The analytic solutions are finally compared with steady `slab jet' adiabatic and radiative numerical simulations, showing a remarkably good agreement between the analytic and numerical results.