The first half was a relatively even affair, but the Blues exploded into life after the interval to put their League Two hosts to the sword.
A brilliant Gareth Evans free-kick got the ball rolling, while Oli Hawkins and Ronan Curtis then struck to effectively wrap up the contest.
Pompey will now have to wait for a week before they learn whether their opponents at the national stadium next month will be Bristol Rovers or Sunderland.
Kenny Jackett made three changes from the side that drew with Barnsley in the league at the weekend.
Two of them were enforced, as Hawkins and skipper Evans replaced the injured Omar Bogle and cup-tied Bryn Morris respectively.
Nathan Thompson had also fully recovered from a knee problem, so Anton Walkes made way at right-back.
Christian Burgess was presented with the Player of the Round award for his quarter-final display before kick-off and made his 150th appearance for the club.
It was an open start to proceedings, with crosses from Byron Moore and Lee Brown causing problems at either end.
The hosts had a great chance to break the deadlock after just five minutes, with Craig MacGillivray getting down to his left to keep out Nicky Maynard’s curler.
But the danger was not over and Brown was forced to hack the ball to safety before a white-shirted player could capitalise.
Pompey then created an opportunity of their own just seconds later, as Curtis spun on the edge of the box and forced Joe Murphy to make a save at his near post.
Hawkins also had an effort blocked by Will Aimson from Brown’s cut-back, with both sides continuing to play with an attacking mindset.
It was Bury’s turn to try their luck on 18 minutes, but former Fratton loanee Caolan Lavery bent his shot narrowly past the post.
The hosts started to see more of the ball as the first half progressed, although were unable to cause MacGillivray any more problems.
Half-chances were the best the Blues could muster at the other end and Curtis drilled wide after Thompson’s teasing delivery was headed away.
And when Viv Solomon-Otabor’s path to goal was blocked, Brown crossed for Hawkins to head narrowly over.
A deep free-kick from Evans caused alarm for the Shakers on 36 minutes, but Curtis could not make a clean connection in the box.
Then, just before the break, Hawkins slipped a pass through to Curtis down the right and his centre had to be headed away by Aimson before it could reach Solomon-Otabor.
Solomon-Otabor did not emerge for the start of the second half, with Jamal Lowe replacing him and Curtis switching to the left.
Pompey looked bright once the action restarted, although a deep free-kick from Evans was dealt with by the Bury back-line.
And when the ball bounced up for Ben Close just inside the area, Aimson bravely dived in to block the midfielder’s volley.
The Blues were dominating possession and long-range attempts by Lowe and Burgess forced Murphy to make saves, albeit comfortable ones.
There was a scare at the other end when Burgess’ ball across goal was almost intercepted by Lavery, but MacGillivray got the ball clear.
Instead it was the visitors who made their pressure count with two goals in quick succession just past the hour mark.
Close was brought down by Callum McFadezen just outside the box and four players gathered around the ball.
But it was EVANS who eventually took the free-kick and bent home a fine effort that left Murphy rooted to the spot.
And the advantage was soon doubled, as Evans’ cross was nodded back across goal by Thompson for HAWKINS to slam into the roof of the net.
Bury soon had chances to reduce their deficit, as Maynard arrowed a shot agonisingly past the post, while Jay O’Shea hit a fierce effort straight at MacGillivray.
But Pompey were still looking to attack themselves and a long-range drive from Evans was parried away by Murphy, with a brilliant Aimson challenge stopping Curtis converting the loose ball.
CURTIS did have his goal on 77 minutes, however, with a fine finish that was lashed right into the side of the net.
And MacGillivray then ensured the visitors’ kept their three-goal lead, diving to his left to keep out a Lavery free-kick that appeared destined for the top corner.
But there was a blow for the Blues late on when Burgess limped off the pitch, with Walkes replacing him for the final few minutes.
Bury (3-5-2): Murphy; Aimson, Thompson, McFadzean (Omotayo 77); Adams (c) (Cooney 70), O’Shea, Rossiter, Mayor, Moore; Maynard (Telford 70), Lavery
Subs not used: Moloney, O’Connell, Stokes, Danns
Pompey (4-2-3-1): MacGillivray; Thompson, Burgess (Walkes 88), Clarke, Brown; Naylor, Close; Curtis, Evans (c), Solomon-Otabor (Lowe 46); Hawkins
Goals: Evans 61, Hawkins 64, Curtis 77
Subs not used: Bass, Haunstrup, Donohue, Vaughan, Pitman
Referee: Ross Joyce
Attendance: 3,900 (452 Pompey fans)