James Tomkins rescued the Hammers with a 90th-minute strike to give Slaven Bilic’s men a 1-0 win in the first leg at Upton Park last week.
The Premier League side were expected to brush the Maltese Cup winners aside, but West Ham will now have to be on their guard to ensure they make it through to the third and final qualifying round.
West Ham could meet Scottish Premiership side Inverness at the next stage after they were drawn together, but John Hughes’ men will have to overturn a 1-0 deficit to Astra Giurgiu to make it through.
After the Hammers’ win last week, Bilic admitted he was relieved Tomkins gave them a first-leg lead but did say he was pleased with his side’s display.
“It is more relief because we scored at the end, but I think we played well, especially the first half,” he said.
“We didn’t play really quick, but quick enough against this type of opposition. They are a decent side, they are not, with all respect, Real Madrid, but they are not like the team from Andorra or San Marino. They can make your job more difficult.”
The Croatian later admitted it would be both an adventure and a challenge if they progressed in the competition.
“The Europa League is great for the club, you could see with the spectators, they are enjoying it, against people from Andorra and Malta – they all love to come for a European night,” he said.
“It is great for a team – it gains not only confidence, but they are different types of games, so it is good for us and for me.
“But on the other hand, it drains you.
“I did it a few times last season, when it was great for Besiktas, but it was also very difficult to play on Thursday in Liverpool, then on the Sunday you have an away game in Turkey.”
Dimitri Payet impressed on his West Ham debut as he scored twice against Southend at the weekend, but he is unavailable on Thursday due to suspension.
Diafra Sakho is also banned from playing in their second-leg clash, but Bilic is still expected to name a strong line-up in Malta.
Mauro Zarate and Modibo Maiga led the line for the Hammers in the first leg, while Mark Noble and Kevin Nolan started in midfield, and all four could feature again, with Adrian expected to be in goal once more.
West Ham have won three European games in a row – after failing to win any of the previous four.
Birkirkara failed to register a single shot on target in the first leg.
English sides have won six of the seven meetings against Maltese clubs – drawing the other – scoring 26 goals and conceding just one.
West Ham are 12/1 to be left to rue their disappointing first-leg showing and be knocked out of the Europa League, with Birkirkara up against substantially shorter odds for the return encounter. Sky Bet offered 50/1 on the Maltese outfit winning at Upton Park but go 7/1 for the minnows to triumph on home turf, while the Hammers are 1/3 for the win. Slaven Bilic’s men are considered 50/1 chances to win Europe’s second-tier tournament, with fellow Premier League outfits Tottenham, Liverpool (both 16/1) and Southampton (25/1) all considered more realistic contenders.