A containment vessel is moved proximate to and preferably surrounding the wellhead, such that leaking oil enters an interior chamber of the vessel. A fluid pump of the containment system is actuated to flow leaking hydrocarbon fluid through a conduit toward a collection sump. A controller controls the speed or volume of a fluid pump to maintain suction force within the interior chamber, or the pressure at wellhead annulus site (PWAS) at a set point that is based on seabed pressure.