We present new ALMA observations tracing the morphology and velocity structure of the molecular gas in the central galaxy of the cluster Abell 1795. The molecular gas lies in two filaments that extend 5-7 kpc to the N and S from the nucleus and project exclusively around the outer edges of two inner radio bubbles. Radio jets launched by the central active galactic nucleus have inflated bubbles filled with relativistic plasma into the hot atmosphere surrounding the central galaxy. The N filament has a smoothly increasing velocity gradient along its length from the central galaxy's systemic velocity at the nucleus to -370 km s-1, the average velocity of the surrounding galaxies, at the furthest extent. The S filament has a similarly smooth but shallower velocity gradient and appears to have partially collapsed in a burst of star formation. The close spatial association with the radio lobes, together with the ordered velocity gradients and narrow velocity dispersions, shows that the molecular filaments are gas flows entrained by the expanding radio bubbles. Assuming a Galactic XCO factor, the total molecular gas mass is 3.2 ± 0.2 × 109 Mo˙. More than half lies above the N radio bubble. Lifting the molecular clouds appears to require an infeasibly efficient coupling between the molecular gas and the radio bubble. The energy required also exceeds the mechanical power of the N radio bubble by a factor of 2. Stimulated feedback, where the radio bubbles lift low-entropy X-ray gas that becomes thermally unstable and rapidly cools in situ, provides a plausible model. Multiple generations of radio bubbles are required to lift this substantial gas mass. The close morphological association then indicates that the cold gas either moulds the newly expanding bubbles or is itself pushed aside and shaped as they inflate.