Slaven Bilic's men were made to work hard for their second-qualifying round win, however, after being taken all the way to penalties at the end of a match they lost 1-0 on the night.
The Maltese minnows made a strong start to things as they looked to overturn their slender first-leg deficit, with a cagey visiting side having to hold their ground in the early stages.
That soon changed, though, as Fabrizio 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 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.
Bilic opted to change things around slightly by moving Kouyate into defence alongside James Collins, as a United side already without Diafra Sakho, Dimitri Payet, Winston Reid and Enner Valencia lumbered on.
Home supporters were out of their seats when Miccoli appeared to be through on goal 10 minutes after the restart, but Collins managed to get across to cut out the opening.
West Ham had weak shouts for a penalty turned down when Jarvis went to ground under the challenge of Mauricio Mazzetti, while up the other end Camenzuli blasted over a decent chance for the Stripes.
Despite being overwhelming underdogs, it was the side from Malta who looked the more lively as the contest edged towards the end of normal time.
There was to be no decisive goal at the end of 180 minutes' play, though, ensuring that extra time and potentially penalties would be required to decide the outcome in the humid conditions.
Jarvis saw his shot deflected wide in a half of limited chances, before the complexion of the contest changed once more when Mazzetti - booked for his part in Tomkins's earlier dismissal - was shown a second booking to earn a red of his own.
Despite the setback, Birkirkara still looked to attack and they gave West Ham a scare when mounting a strong attack that Adrian was alert to.
Giovanni Tedesco's charges soon reverted into a solid defensive set-up in the dying stages, however, with the visitors unable to find a breakthrough to pave the way for penalties.
Having found the net with their first three spot kicks, it was advantage West Ham when Nikola Vukanac blasted high over the bar.
It was then left to Diego Poyet to win it for the English outfit, and he did exactly that by smashing the ball home to set up a meeting with Romanian side Astra Giurrgiu in the next round.