Csaba Laszlo admitted Hearts rode their luck as they stunned Celtic to reach the Co-operative Insurance Cup semi-finals.
The Jambos won courtesy of a Michael Stewart penalty but the holders, who had Stephen McManus sent off for a wild lunge in stoppage time, missed a succession of gilt-edged chances to turn the tie around.
“This was a very important win. We had also a little bit of luck,” said Hearts boss Laszlo, who felt his centre-backs escaped with a clean sheet.
“I think the communication between Ismael (Bouzid) and Jose (Goncalves) was not the best today. But in football, you must score goals and we scored one more.
“Last season, we flew very early out of the cup. Now we've come to the best four.
“I hope in the draw we'll have a bit of an easier opponent.”
While St Mirren, St Johnstone and Rangers complete the semi final draw, Hoops manager Tony Mowbray will have to find answers to his side's shocking home form, with two wins out of nine at Celtic Park since taking over from Gordon Strachan in the summer.
With only around 20,000 inside the stadium at kick-off, and the top tier closed altogether, there was a ghostly atmosphere as Hearts got the game going.
The home side almost went a goal behind within a minute when Christian Nade’s shot was headed off the line by Andreas Hinkel.
The much-changed home side quickly took a grip of the game in terms of possession but there was enough unease in their defence to suggest the Edinburgh side could find some joy.
Hoops' cult hero Paddy McCourt replaced Maloney for the start of the second half.
Within six minutes the Northern Ireland international produced a moment of magic, leaving Hearts right back Craig Thomson for dead 40 yards from goal, drove inward at goal and cracked the bar with a thundering drive from the edge of the box.
But in the 58th minute the Celtic fans were silenced when the visitors took the lead.
Fox tripped Nade inside the box as the Frenchman drove in at goal and although Zaluska got his hand to Stewart's unconvincing penalty he failed to stop the ball making its way over the line.
Celtic’s response was impressive with Samaras having a close-range shot brilliantly blocked by Balogh and seconds later the Greece international chipped the Hearts keeper from 14 yards only to see the ball strike the base of the post and bounce out.
In the second minute of added time McManus was red carded for a wild lunge at Driver, referee Thomson upgrading it from yellow on the advice of his linesman.