Roared on by a passionate, near-capacity crowd in Tynecastle for the first time in continental competition in nearly six years, captain Michael Stewart sparked dreams of overturning the 4-0 first-leg deficit with his 16th-minute opener.
The midfielder then struck the crossbar as Csaba Laszlo's outfit made a mockery of Dinamo Zagreb's dominant display last week.
When defender Marius Zaliukas hooked in the second shortly after the break, belief they were capable of a sensational result coursed through the home ranks.
However, the tie, and the club's chances of reaching the lucrative group stages of the Europa League, was ultimately lost in the horribly below-par performance in Croatia.
The game survived a pre-match scare when a fire caused by one of the back-up electricity generators blew one set of floodlights at the Gorgie Road end of the ground.
However, German referee Knut Kircher ruled the tie could go ahead minus one set of lights.
Having been told he cannot sign the striker he so desperately wants, Laszlo gave Christian Nade and Gary Glen the opportunity to impress, but it was the return from suspension of Zaliukas and Stewart from Sunday's 2-1 defeat to Rangers that brought the goals.
Hearts sought the explosive start they needed right from kick-off and following three long throws from Jose Goncalves down the left David Obua's mis-kicked his shot when well placed.
Zagreb should have taken the lead on the night moments later, only for Greece internationalist Dimitrios Papadopoulos to whistle his shot inches wide from the right side of the box.
A wonderful piece of skill by Hearts winger Suso Santana along the by-line in the 11th minute failed to bring its rewards, despite goalkeeper Tomislav Butina's nervous fumble of the Spaniard's cross as Hearts continued to press.
Six minutes later, Glen got a clear sight of goal from Obua's driven pass but the youngster's first touch let him down and he could not get a clean shot away.
However, Stweart ignited the match by getting the early goal Hearts craved.
The Scotland international robbed Milan Badelj 40 yards from goal and burst past Ivica Vrdoljak before slotting past Butina.
The Croatian champions attempted to cancel out Hearts' advantage immediately but Janos Balogh was equal to Marijo Mandzukic's shot from Papadopoulos' cutback.
Stewart almost got a dream second in the 28th minute as the woodwork denied Hearts.
Craig Thomson's deep cross was knocked back in by Obua and into the path of Stewart by Glen but the skipper's hooked volley took a slight deflection before rebounding back off the crossbar.
Hearts were temporarily down to 10 men at the start of the second period after Zaliukas took a kick to his face and suffered a cut mouth.
However, the Lithuanian was able to continue and hauled the capital side even closer to a dramatic comeback in the 55th minute.
Thomson's corner was only half-cleared to the edge of the box and Zaliukas hooked the ball back through a crowded goalmouth and past Butina.
Immediately, Papadopoulos should have taken the wind out of Hearts but shot over from the edge of the box and Badelj's powerful drive went the same way.
Hearts had claims for a penalty waved away in the 68th minute when Santana went down in the box, and then went close to setting up a grandstand finish with 12 minutes remaining.
However, Butina scrambled Ismael Bouzid's header from Thomson's free-kick away from his near post.
Balogh beat a shot from Papadopoulos away as play raged from end to end before Obua wasted Hearts' last real chance to set up a late rally when he hooked his shot over the bar with seven minutes left.
It was a magnificent performance which was night and day from that in Zagreb and one which went some way to restoring the reputation of Scottish football in European competition.