Steven Fletcher missed a great chance for the home side early on and was then sent off on the half-hour mark for a heavy challenge on Hearts skipper Christophe Berra in the fourth-round tie at Easter Road.
Hibs were further punished seven minutes from the interval when Nade knocked in an Andrew Driver cut-back from close range.
The visitors controlled the second half comfortably and in the third minute of injury time Glen, on for David Obua, knocked in number two after running on to a fine pass from Christos Karipidis.
Hibs will point to Fletcher's dismissal as the turning point but the fact remains that the Leith club are still waiting to end their 107-year wait to win the Scottish Cup.
The Hibs fans turned up for this lunchtime kick-off in optimistic mood although Easter Road was not full by the time the home side set about clinching a fifth-round spot.
The match quickly descended in to a scrappy affair but in the 11th minute Hibs should have taken the lead after midfielder Souleymane Bamba slipped in Fletcher with a pass which split the square Hearts defence open.
Fletcher tried to go around Janos Balogh but the Jambos goalkeeper guessed right and managed to take the ball from his feet and the Hibs striker was then penalised for a foul on Eggert Jonsson as he tried to win back the loose ball.
It was the first frustrating moment for the Scotland striker, who would not see out the opening 45 minutes.
As on-field temperatures increased, Berra was booked for a foul on Jonatan Johansson but the Hearts skipper was the victim when Fletcher was red-carded on the half-hour mark.
The Hibs striker lost possession from a long Lewis Stevenson pass and in his frustration he left his foot in against the Hearts skipper, leaving referee Craig Thomson with little option but to send him off.
The dismissal fired up the home fans, who viewed the decision as an injustice, but for the first time in the game Hearts sensed they had the advantage and in the 38th minute Nade fired them ahead.
When Hibs failed to clear another attack, Driver found himself with space on the left-hand side and goalkeeper Yves Ma-Kalambay ambling towards him leaving his goal exposed.
When the Hearts midfielder cut the ball across the six-yard box Nade pounced ahead of Hibs skipper Rob Jones to knock the ball high into the net.
The cup tie had swung quite clearly towards the Tynecastle side and there appeared to be more good news for the visitors when Ma-Kalambay was left in the dressing room at the interval with a reported hamstring injury.
Former Dundee United goalkeeper Grzegorz Szamotulski, who signed until the end of the season on January 2, took over to make his Hibs debut.
Szamotulski almost had a nightmare start when he struggled with a David van Zanten back-pass but while he escaped, Hearts' increased energy and enthusiasm promised a difficult Easter Road baptism for the Pole.
The traffic continued towards the Hibernian goal, the home side finding little respite in the lashing rain as maroon jerseys came at them from both flanks and through the middle.
Hibs brought on Colin Nish and Alan O'Brien for Johansson and Derek Riordan in the 65th minute with Riordan being shown a yellow card on his way off for gesturing to the Hearts supporters.
The double substitution gave the home side renewed enthusiasm as they went all out for the equaliser but Hibs boss Mixu Paatelainen irked referee Thomson so much with his complaints that he was reprimanded.
However, in going forward, the Leith side were leaving gaps and they were let off the hook in the 77th minute when Nade's header from Driver's cross was saved comfortably by Szamotulski, the goalkeeper making a better save from Driver with four minutes remaining.
In the third minute of injury time Hearts substitute Glen made sure of a place in the fifth round when he took a pass from Karipidis before rounding Szamotulski and knocking the ball into the empty net.