Gary Mackay, Craig Levein, Ian Baird and Glyn Snodin, who notched his first for the club, scored the goals which swept Slavia out.
The visitors, who held a 1-0 lead from the first leg, showed their dark side when Penicka was ordered off in the second half.
Hearts played some of their sweetest football of the season in the first half but lapses in concentration almost cost them dear, particularly when ten-man Slavia got a second goal which would have taken them through had Snodin not scored a spectacular winner in the 79th minute.
Hearts started on the front foot and took the lead after ten minutes when Mackay started the move in his own half with a raking pass to Baird. He laid it off to Snodin, whose angled through ball encouraged Robertson to shrug off his marker. The striker fed the ball back to Mackay who cracked a 16-yard shot past Janos into the corner of the net.
Hearts lost their concentration and Slavia were level in the 14th minute. Necas provided the cross from a corner and Silhavy stabbed the ball over the line. It was a huge blow to Hearts’ hopes to concede the away goal and the loss meant they had to win by two clear goals.
They set about the task the right way, though, by regaining the lead in the 21st minute. Bannon made a surging run down the right, the ball was flighted beautifully to the near post where Ian Baird rose to send a header soaring past Janos.
Hearts finished the first half on a high note by taking the outright lead in the tie for the first time with a simple goal which came from a set-piece. Bannon supplied the corner and Levein headed the ball into the corner.
The Czech side’s chances were reduced when Penicka was sent off in the 58th minute for a foul on Baird.
Despite being reduced to ten men Slavia remained dangerous and scored a second goal in the 65th minute. Necas opened up the Hearts defence and Kuka lobbed the ball brilliantly over the advancing Smith.
The drama was not finished however, as Hearts secured their place in the second round with a superb goal 11 minutes from the end.
The Jambos were awarded a free kick 25 yards from goal and Snodin sent a magnificent shot curling round the wall into the top right-hand corner of the net.