Match ends, Dundee 3, Falkirk 1.
Two goals from Kane Hemmings and one from Gary Harkins were enough for Dundee to eliminate last season's Scottish Cup finalists, Falkirk.
The Bairns had the better of the first half, but Hemmings hammered in a stunning strike after 39 minutes.
Falkirk were level within two minutes when Paul Watson profited from poor home defending.
But Hemmings converted Paul McGinn's cross to make it 2-1 on the hour before turning provider for captain Harkins.
Dundee's impressive second-half display means that Paul Hartley's side secure a fifth-round tie away to Dumbarton.
Falkirk, who are third in the Championship - 27 points ahead of the Sons in seventh - started more positively than their Premiership opponents with a high-tempo pressing game that didn't allow Dundee to settle into any sort of rhythm.
The Bairns looked slicker when in possession, with Craig Sibbald finding plenty of space going forward down the right side.
It was Sibbald's set-piece delivery that created an early chance for the visitors, but Aaron Muirhead steered the dangerous corner wide from eight yards.
Dundee had a strong wind at their backs in the opening period, but they were guilty of over-hitting forward passes as they failed to make the most of the advantage.
Harkins did provide a rare flash of brilliance after 18 minutes with a lovely turn on the edge of the box before a delicate chip sent Nick Ross clear.
The midfielder, part of the Inverness Caledonian Thistle side that won the Scottish Cup against Falkirk in May, lashed his shot high and wide.
Hemmings showed him how to do it when Dundee broke the deadlock in controversial fashion six minutes before the interval, with Peter Houston's side furiously claiming for offside.
The striker did not hang about to debate it as he drilled a stunning drive from the right edge of the area in off the underside of the crossbar.
The lead lasted just two minutes and, once again, it was some dodgy defending that proved costly for Hartley's side.
Kostadin Gadzhalov was at fault, staying well behind the rest of the backline claiming an injury, only to play John Baird onside.
The striker made the most of the opening to drive the ball across the six-yard box, where central defender Watson forced it over the line.
Dundee were far more impressive in the second half and Hemmings claimed his second of the night with another smart finish after 61 minutes.
McGinn set it up with a perfectly-flighted deep cross on the run down the right and Hemmings hammered the ball into the net from close range for his ninth goal in his past six games and 15th of the season overall.
Harkins ended any doubt about the outcome in the 73rd minute with a terrific goal on the counter-attack as Falkirk were caught trying to find a way back into the tie.
Hemmings played a crucial part with the pass that his captain guided into the corner of the net from the edge of the area.
It was a first defeat in 14 games for Falkirk, who will have to pick themselves up for a crucial league match against Rangers on Saturday.
- 26GadzhalovSubstituted forEtxabegurenat 76'minutes
- 8LowSubstituted forLoyat 71'minutes
- 44WatsonBooked at 55mins
- 7TaiwoSubstituted forMcHughat 67'minutes
- William Collum
- Shots on Target
Second Half ends, Dundee 3, Falkirk 1.
Julen Etxabeguren Leanizbarrutia (Dundee) wins a free kick in the attacking half.
Foul by John Baird (Falkirk).
Attempt missed. Blair Alston (Falkirk) right footed shot from outside the box misses to the left.
Corner, Falkirk. Conceded by Scott Bain.
Attempt saved. Will Vaulks (Falkirk) right footed shot from a difficult angle on the right is saved in the centre of the goal.
Foul by Kevin Holt (Dundee).
Aaron Muirhead (Falkirk) wins a free kick in the defensive half.
Foul by Paul McGinn (Dundee).
David Smith (Falkirk) wins a free kick in the defensive half.
Corner, Falkirk. Conceded by Paul McGowan.
Greg Stewart (Dundee) hits the right post with a right footed shot from the right side of the box.
Attempt saved. Blair Alston (Falkirk) header from the centre of the box is saved in the centre of the goal.
Substitution, Dundee. Julen Etxabeguren Leanizbarrutia replaces Kostadin Gadzhalov because of an injury.
Goal! Dundee 3, Falkirk 1. Gary Harkins (Dundee) right footed shot from the centre of the box to the bottom left corner. Assisted by Kane Hemmings.
Attempt missed. Paul McGowan (Dundee) right footed shot from outside the box is high and wide to the right.
Substitution, Dundee. Rory Loy replaces Nicky Low.
Corner, Falkirk. Conceded by Kostadin Gadzhalov.
Substitution, Falkirk. Robert McHugh replaces Tom Taiwo.