The game had been postponed just over a couple of weeks earlier due to some heavy rain in Cheshire.
But the Blues ensured the rearranged contest was no damp squib, with George Hirst’s early header releasing the floodgates.
Tyler Walker soon opened his account for the club and there were plenty more opportunities to score before Hirst wrapped things up after the break.
The Railwaymen kept battling away, but Chris Long’s late strike proved no more than a consolation for League One’s bottom side.
Danny Cowley made three changes from the side that thrashed Accrington on home soil at the weekend.
Ronan Curtis returned from a two-match suspension, while there were also starts for Joe Morrell and Walker.
Marcus Harness missed out as he begins his own ban following a red card against Stanley, while Ryan Tunnicliffe and Aiden O’Brien both dropped to the bench.
Pompey were unsurprisingly playing with plenty of confidence as they looked to extend their fine run of recent form.
And there were not even three minutes on the clock when they broke the deadlock after Morrell had stolen the ball from Regan Griffiths in the middle of the pitch.
He worked the ball to Curtis, who in turn found Connor Ogilvie on the left and his cross was firmly headed home by the leaping HIRST.
They had an excellent chance to double their lead following a clever free-kick routine that saw Ogilvie leap over the ball before Curtis hit a shot.
His low effort could only be parried out by Dave Richards, but the keeper recovered quickly to deny Walker from the rebound.
The striker was then teed-up by Morrell, spun sharply in the area and saw his effort deflected past the post.
But WALKER did have his first Blues goal on 31 minutes, following up to slot home after Sean Raggett had taken advantage of some sloppy defending to fire in a shot that was saved.
The game began to get bogged down by a series of niggly challenges that halted any rhythm built up by either side.
It was the visitors who finished the half on top, though, and they had a couple of opportunities to extend their advantage further in stoppage-time.
Curtis was first up, holding off the challenge of Dan Agyei and cutting inside before drilling the ball narrowly past the post.
Some patient build-up play then led to Ogilvie whipping in a cross and Hirst played a pass through to Louis Thompson, who could not beat Richards from an acute angle.
Crewe Alexandra 0
Pompey’s pressing caused more problems when the action restarted and Mahlon Romeo stole possession.
Hirst then played the ball to Walker, but the striker could not quite keep his shot low enough to test Richards.
They were also showing plenty of determination at the other end, with Ogilvie flinging himself in the path of two Agyei efforts in quick succession.
It was only Richards preventing Crewe from falling further behind as the second half progressed, with the keeper making a couple of superb stops.
The first came when Ogilvie brilliantly brought down a Curtis free-kick and hit a volleyed effort that appeared destined for the neat.
Then, shortly afterwards, Richards did well to repel a neat Walker back-flick from a low Ogilvie delivery.
It seemed like only a matter of time before the Blues scored again – and a third did arrive on 66 minutes.
Just seconds after seeing a shot saved from Richards, HIRST met an Ogilvie cross with a downwards header that bounced up and into the roof of the net.
Cowley began to turn his attention towards the weekend, with Thompson and Hirst withdrawn for Tunnicliffe and O’Brien.
Crewe almost grabbed a late lifeline, but a back-pedalling Gavin Bazunu managed to tip over a Zac Williams curler.
They went even closer to scoring when a corner led to a spot of pinball in the Blues box and Connor O’Riordan stabbed the ball against the crossbar.
And Long denied the visitors a clean sheet in stoppage-time, when he managed to sneak a shot inside Bauznu’s near post.
Crewe (3-4-1-2): Richards; Johnson (Williams 46), O’Riordan, Offord; Agyei (Ainley 67), Murphy (c), Griffiths (Harper 46), Lundstram; Lowery; Long, Porter
Goals: Long 90+1
Subs not used: Jaaskelainen, Finney, Sambou, Salisbury
Pompey (3-4-1-2): Bazunu; Carter (Hume 78), Raggett, Robertson (c); Romeo, Morrell, Thompson (Tunnicliffe 68), Ogilvie; Curtis; Hirst (O’Brien 71), Walker
Goals: Hirst 3, 66, Walker 31
Booked: Thompson, Carter
Subs not used: Webber, Vincent, Jewitt-White, Mingi
Referee: Andrew Kitchen
Attendance: 4,307 (1,050 Pompey fans)