There were early scares, with the goalkeeper Jack Hamilton having to block from close range. Hearts knew patience would be required, but wastefulness was their undoing.
Hearts eventually pulled a goal back when Sammon headed Djoum's cross into the net, but the home side had left themselves too much to do.
Second-half goals by Christian Bubalovic and Edward Herrera - against the run of play - stunned the home crowd.
Hearts had approached the game full of confidence after drawing the away leg 0-0 last week, not least because Birkirkara had won only two of their previous 23 ties in Europe.
Birkirkara is one of the founding members of the European Club Association.
Bubalovic was calm and ruthless when the ball dropped to him inside the penalty area after a free-kick wasn't cleared by the Hearts defence, and he struck a shot high and hard into the corner of the net.
Earlier, Prince Buaben missed a first-half penalty while Jamie Walker and Sam Nicholson both hit the woodwork.
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Hearts were frustrated by their own shortcomings. Despite periods of possession, they were ragged in their passing and could not use their width to good effect against well-organised opponents.
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The same fate befell Nicholson after the break, when he skipped infield on the left and sent a shot crashing onto the bar, with the ball bouncing clear. By then, though, Hearts were behind.
Anxiety now hampered Hearts' play as they urgently tried to rescue the tie, and Arnaud Djoum lacked composure as he steered the rebound over after Juanma's shot was blocked by Kopric.
Conor Sammon's header gave Hearts hope, but they could not find a leveller.
Birkirkara took advantage of that mindset. When one attack broke down, the visitors moved the ball upfield and out to Herrera, who ghosted in from the right of the penalty area and prodded a shot through Hamilton's legs.
Note 1: match forfeited, result set to 0–3 by UEFA.
Hearts were unable to create openings, and even when they did they fell short. Walker met a Callum Paterson cross with an acrobatic volley that sent the ball over his shoulder but onto the underside of the bar.
Nicholson was the home side's brightest player, and he won a penalty kick when he skipped into the box and was brought down by Cain Attard.
As Buaben prepared to take the spot kick, Birkirkara's manager Josef Mansueto shouted to his players that it would go to the goalkeeper's left and Miroslav Kopric duly heeded the advice, diving to that side to save a weak effort.
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Birkirkara Football Club is a football club based in the town of Birkirkara, the largest town in the island of Malta. The club was formed in 1950, following the amalgamation of Birkirkara United and Birkirkara Celtic. Birkirkara currently play in the Maltese Premier League, which it has won on four occasions, most recently in the 2012–13 season. Birkirkara also has a Futsal team under the same name.
The Maltese team were more adventurous than in their performance at home, though, and their ability to break swiftly and sharply on the counter carried them through to face Krasnodar of Russia in the third qualifying round.