One of the major discoveries of Hinodes Extreme-ultraviolet Imaging Spectrometer (EIS) is the presence of upflows at the edges of active regions. As active regions are magnetically connected to the large-scale field of the corona, these upflows are a likely contributor to the global mass cycle in the corona. Here we examine the driving mechanism(s) of the very strong upflows with velocities in excess of 70 km/s, known as blue-wing asymmetries, observed during the eruption of a flux rope in AR 10977 (eruptive flare SOL2007-12-07T04:50). We use Hinode/EIS spectroscopic observations combined with magnetic-field modeling to investigate the possible link between the magnetic topology of the active region and the strong upflows. A Potential Field Source Surface (PFSS) extrapolation of the large-scale field shows a quadrupolar configuration with a separator lying above the flux rope. Field lines formed by induced reconnection along the separator before and during the flux-rope eruption are spatially linked to the strongest blue-wing asymmetries in the upflow regions. The flows are driven by the pressure gradient created when the dense and hot arcade loops of the active region reconnect with the extended and tenuous loops overlying it. In view of the fact that separator reconnection is a specific form of the more general quasi-separatrix (QSL) reconnection, we conclude that the mechanism driving the strongest upflows is, in fact, the same as the one driving the persistent upflows of approx. 10 - 20 km/s observed in all active regions.