The Maltese minnows made a strong start to things, with a cagey visiting side having to hold their ground in the early stages.
That soon changed, though, as Miccoli managed to convert a cross from six yards out to leave the tie delicately poised at the Infetti Ground.
It was West Ham who upped things in search of the second goal of the night, but they were often frustrated by Birkirkara's resolute backline.
Kouyate looked to take things into his own hands by charging 20 yards up the field, only for his shot to easily be collected by Hamer.
Things turned even worse for Slaven Bilic's charges just prior to the interval, when first-leg hero James Tomkins was shown a straight red following a coming together with Nikola Vukanac to leave the Hammers with it all to do.