Captain Alim Ozturk went the closest to a winner but his long-range effort crashed off the crossbar.
Conor Sammon did find the net but the flag was raised for offside.
Callum Paterson twice went close in the second half with a shot and a header, but they could not find a winner.
The game began with the temperature gauge hitting 30C, however Robbie Neilson's side thought they had the perfect start after just three minutes.
Sammon latched onto a through-pass from Prince Buaben before firing the ball across the goalkeeper and into the net only for the far-side assistant to flag for offside.
The home side were awarded a free-kick around 20 yards from goal on the angle and Srdan Dimitrov curled the ball towards goal at pace, but Jack Hamilton did well to save at the second attempt.
Hearts continued to dominate possession and Ozturk floated a free-kick towards the back post, but Miroslav Kopric in the home goal punched the ball clear.
Jamie Walker was teasing the Birkirkara full-back Cain Attard at every opportunity and the defender was caught napping when the Hearts winger ghosted in, but his header was straight at Kopric.
The second period continued in the same vein with Hearts pushing forward at every opportunity.
Patterson forced Kopric into a diving save with a header from a Walker corner.
There were shouts for a Hearts penalty by the travelling fans when Arnaud Djoum tumbled inside the box after a challenge by Emerson, but the Bulgarian referee waved play on.
On a rare venture forward Attard found himself driving towards the Hearts goal but his effort was well wide of Hamilton.
Ozturk is well known for his long-range efforts and only the crossbar denied the Tynecastle defender the opener from 40 yards out.
Another close call for the home side - who were knocked out by West Ham on penalties at this stage last season - came when Paterson volleyed inches wide from 12 yards with the goalkeeper stranded.
Substitute Sam Nicholson provided a chance for the opener when he fired the ball across for Juanma, but the striker took the wrong option and the chance was gone, with the game ending goalless and setting up a Tynecastle thriller next Thursday.
Hearts return to Tynecastle for next week's second leg of their Europa League second round qualifier against Birkirkara on the back of a goalless draw in the Maltese heat.