In front of their biggest home crowd in 23 years, first-half goals from Jason Cummings and Grant Holt put Neil Lennon’s team in command of the replay before substitute Andrew Shinnie added a third.
Hearts pulled one back through Esmael Goncalves, but never really looked capable of stopping the fired-up 2016 Scottish Cup winners marching on to set up a home clash with fellow Championship side Ayr United in the quarter-finals on March 4.
The result means Hibs, who also defeated Hearts in a fifth-round replay at Easter Road a year ago, are now on their longest unbeaten run against their Capital foes since the 1970s after going seven derbies without losing since suffering a 2-1 setback at Tynecastle in August 2014.
Hibs made three changes to the side that incurred the wrath of manager Lennon after drawing 1-1 at Raith Rovers last Saturday. Chris Humphrey, Martin Boyle and Grant Holt came in for James Keatings, Fraser Fyvie and Brian Graham as the hosts lined up in a conventional 4-4-2 formation. There was no place in the squad for Paul Hanlon or Dylan McGeouch.
Hearts made one change from the side that drew 1-1 with Inverness as Malaury Martin replaced Moha Choulay. There was no place in the squad for Don Cowie, who suffered a rib injury in the goalless match between the teams at Tynecastle ten days previously.
Hibs were dealt a blow after just three minutes, however, when Humphrey was forced off by injury and had to be replaced by Shinnie.
Cummings had the first attempt of the match after 12 minutes but his shot from 20 yards was easily gathered by Hearts goalkeeper Jack Hamilton. However, the Hibs striker wasn’t to be denied. With 19 minutes gone, Shinnie seized possession just inside the Hearts half and sliced open the visitors’ defence with a precise through ball. Cummings latched on to it just outside the box and composed himself before slotting a clinical finish past Hamilton from around 15 yards out. The Easter Road support erupted as their top scorer celebrated his fifth goal in six games against Hearts.
The hosts threatened again on the half hour when Cummings whipped in a dangerous inswinging free-kick but Liam Fontaine headed wide after colliding with his defensive colleague Darren McGregor while attacking it at the far post.
Hibs went even closer in the 33rd minute when John McGinn burst into the danger area and saw a goalbound shot brilliantly beaten away by Hamilton before Shinnie fired agonisingly over from the rebound, although it looked like Perry Kitchen had deflected.
The Easter Road side extended their lead three minutes later, however. Cummings was the architect with a delightful through pass for Holt, who slipped it beyond Hamilton. Hearts were in serious bother and their supporters soon started jeering at what they saw as a lacklustre approach to their highest-stakes match of the season.
Hearts head coach Ian Cathro made a double substitution for the start of the second half, with captain Kitchen and Martin making way for attacking pair Sam Nicholson and Rory Currie.
Hibs almost struck for a third time in the 49th minute when McGinn burst into the box and cut the ball back for Shinnie, but Hamilton stood up to make an excellent instinctive block to deny the on-loan Birmingham City midfielder.
Hearts finally came to life as an attacking force in the 58th minute when they worked their way into a dangerous position and Walker flashed a powerful shot just over from a Nicholson cross.
But they soon found themselves staring at a thrashing as Shinnie’s shot from 20 yards out swerved beyond Hamilton, leaving the home support in a state of delirium.
The visitors grabbed themselves a lifeline in the 69th minute when they won a penalty after Goncalves was tripped by McGregor as he burst through on goal. Goncalves’ spot-kick was saved by Ofir Marciano, but the Hearts striker forced the ball over the line at the third attempt.
Hibs thought they had restored their three-goal advantage when McGregor headed in a Cummings free-kick in the 78th minute but the linesman’s flag cut short their celebrations. Aside from a couple of strikes from Arnaud Djoum and Nicholson, Hearts rarely looked like making the closing stages particularly uncomfortable for their hosts, who were able to savour victory long before the end.
Hibs: Marciano, Gray, Fontaine, McGregor, Stevenson, Bartley, McGinn, Humphrey, Holt (Graham 75), Cummings (Fyvie 87), McGregor. Subs: Laidlaw, Keatings, Martin, Donaldson.
Hearts: Hamilton, Struna (Choulay 75), Hughes, Avlonitis, Sowah, Kitchen (Nicholson 46), Tziolis, Martin (Currie 46), Djoum, Walker, Goncalves. Subs: Noring, Rherras, Nowak, Johnsen.
Referee: Steven McLean