Sean Clare's second-half penalty earned Hearts passage to the Scottish Cup quarter-finals after a hard-fought 1-0 win over League One Falkirk.
Clare punished the home side after Michael Doyle was adjudged to have pushed over Liam Boyce five minutes into the second half.
In dreadful conditions at the Falkirk Stadium, the Bairns hit the woodwork three times as they sought an equaliser but the Ladbrokes Premiership's bottom side weathered the storm to reach the last eight.
Hearts came into the tie having conceded six goals in their last two games and they were forced into a defensive change due to the absence of John Souttar because of an ankle injury.
In came Ben Garuccio, Conor Washington, Clevid Dikamona and Michael Smith as manager Daniel Stendel switched to a three-man rearguard.
The Tynecastle side were massive favourites against their League One hosts but, sitting bottom of the Premiership, there would have been doubts as they battled their way through Storm Ciara to the Falkirk Stadium.
Hearts attempted to defeat the conditions by getting their passing game early on but the hosts were keen not to allow their visitors to settle.
The Jambos twice worked themselves into space on the left flank in the opening exchanges. Ben Garuccio's driven cross was missed by Falkirk skipper Gregor Buchanan six yards out but there was no-one in maroon in the middle to capitalise.
The Gorgie outfit also had two penalty appeals waved aside by referee John Beaton on a day when the weather meant set-pieces could prove vital.
Skipper Steven Naismith failed to convince when he fell attempting to collect a low corner and moments later Clare went down in the area under pressure from Buchanan and Falkirk keeper Robbie Mutch.
Falkirk attempted to hit on the break but Craig Halkett was alert to snuff out the danger as Conor Sammon and Josh Todd lurked.
And the home side passed up a gilt-edged chance to break the deadlock seven minutes from the interval. A long kick-out from Mutch was missed by Dikamona and seized on by Josh Todd but the winger sliced wide on the angle.
Then, Hearts sprung to life with their best opportunity of the half. Liam Boyce engineered some space for a shot but Mutch dived low to his left to hold on.
The match then suddenly burst to life within minutes of the break as Hearts took the lead.
A long punt from keeper Joel Pereira sailed over the Falkirk defence and found Boyce running free. Michael Doyle's tug was punished by Beaton pointing to the spot and Clare hammered in the penalty.
Falkirk then sought an immediate response and hit the frame of the goal three times in the space of six minutes.
First, Declan McManus stopped a header from Buchanan on the Hearts line and his shot on the turn crashed back off the post.
Moments later, the striker drilled a low shot off the same upright after being found in space on the right side of the box.
Then, midfielder Gary Miller came within inches of the equaliser with a drive from edge of the area that came back off the crossbar.
Hearts substitute Euan Henderson should have sealed the victory six minutes from time but clipped the top of the bar at the other end from just seven yards out.
