Can the Moon provide momentum to an object in Earth's orbit? Gravity Assisted Boost
The Moon has a "Roche lobe", where the Moon's gravity dominates, and all you need for the situation you describe is that the Roche lobe lies physically outside the Moon.  That will be true, because the Lagrange points are on the outside edge of that Roche lobe, and it's outside the Moon.  The Moon's Roche lobe is shown here: http://hyperphysics.phy-astr.gsu.edu/hbase/Mechanics/lagpt.html