Heart Of Midlothian 1 Kilmarnock 0
Jankauskas' goal came from a great diagonal ball by Rudi Skacel that cut out the defence and left the big Lithuanian to side-foot past the on rushing keeper and into the net.
In truth though, the game never really came alive with too many niggly fouls to allow for a flowing encounter with Kilmarnock the main perpetrators.
The fist half never took off until Jankauskas' 35th minute strike but up until then the best chances of the half had come from an unlikely source.
Twice Jamie McAllister tried the low shot from the edge of the box but twice he was denied by Graham Smith in the Killie goal.
Kilmarnock's only chance of note was sent wide by Kris Boyd after a sharp turn and shot.
The second half kicked off with one change - referee Stuart Dougal was replaced by the fourth official Stevie O'Reilly.
The second half followed much the same path as the first and again it was Hearts who had the best chances.
The first fell to Jankauskas who should have done better than head McAllister's cross over the bar.
Fifteen minutes into the half Kilmarnock had what appeared to be a strong appeal for a penalty. With he ball bouncing around the box, Danny Invincibile struck it goalbound only to see it hit the outstretched arm of the defender but the referee waved away the appeals.
Hearts could have tied the game up in injury time but McAllister passed wide with the goal gaping.
It wasn't needed though and Hearts held on without seriously being threatened by Killie.