Substitute Miroslav Matusovic scored a late goal to add to Daniel Kolar's first-half opener to surely make the second leg in the Czech Republic a formality.
The scoreline did not flatter Sparta Prague as Hearts failed to put the Czech side under any sustained pressure.
Hearts were hit by injuries and suspensions and Mirsad Beslija came into the midfield and youngster Jamie Mole was also included.
Sparta looked more dangerous throughout and Ludovic Sylvestre slid a shot narrowly wide early on after exchanging passes with Libor Dosek.
Karol Kisel then shot wildly over the top after another dangerous Czech move had Hearts on the back foot.
Craig Gordon was the busier of the two goalkeepers and he did brilliantly to touch a shot from Sylvestre over the bar for a corner.
It was no surprise when Kolar beat the Scotland goalkeeper with a well-placed right-foot shot from the edge of the box which flew high into the net after 33 minutes.
Hearts were in further trouble when Jiri Homola had a header from just two yards out from a Jan Simak corner but sent his effort over the crossbar when he should have scored.
The home side had a chance to equalize moments before half-time but Mauricio Pinilla sent his header straight into the arms of Jaromir Blazek from a Paul Hartley corner.
Hearts started the second half better and Mole was denied by Blazek after 58 minutes after working himself an opportunity.
Roman Bednar then tried his luck from 25 yards but his effort curled wide of the post.
But Sparta were always dangerous on the break and substitute Matusovic should have done better with an effort from six yards out which he sent wide.
Gordon made superb stops in quick succession from Libor Dosek and then Kisel to keep Hearts hanging on.
But the goal was only delayed until the 71st minute when Matusovic took a pass from Kisel and fired a low shot into the corner of the net to kill off the home side.