Goals from Paul Kalambayi and Aaron Pressley put AFC Wimbledon into a commanding position at Plough Lane.
But Harrison netted either side of the break to level the contest and then completed his treble to turn the game on its head.
Pressley then equalised from the spot and a penalty shoot-out was looming before Will Nightingale headed the hosts back in front.
The Blues poured forward in search of another goal, but were caught on the break as Luke McCormick wrapped things up in stoppage-time.
Danny Cowley changed the entire starting line-up from Pompey’s last outing at Wigan 10 days earlier.
There were three debuts handed out, including keeper Jake Eastwood following his emergency loan switch from Sheffield United.
Summer signing Louis Thompson also came into the midfield, while deadline day capture Mahlon Romeo was at right-back.
The defence was completed by Haji Mnoga – who can feature in this competition despite his loan move to Bromley – Connor Ogilvie and Paul Downing.
Michael Jacobs, Reeco Hackett-Fairchild, Gassan Ahadme, Harrison and George Hirst completed the team, with the latter making his first start for the club.
There was a youthful look to the bench, with the seven substitutes all members of Liam Daish’s under-18 side.
The visitors got off to the worst possible start and found themselves trailing just 61 seconds into the contest.
A corner found its way to Kalambayi inside the area and the defender had far too much time to find the net.
The Blues responded well to that setback, however, and started to look a real threat going forward themselves.
A low shot from Gassan was kept out by Zaki Oualah, while the keeper soon flapped at a Jacobs free-kick and had to be bailed out by his defence.
Romeo then raced onto a pass over the top by Thompson, but could not beat the Dons stopper from an acute angle.
Pompey looked nervy at the other end, though, and some sloppy play provided an opportunity for the home side.
But Ayoub Assal and Pressley both took too long over deciding to shoot and the latter’s eventual effort was deflected wide.
Neither side seemed comfortable at the back, although Kalambayi made a vital block to deny Harrison.
It was Wimbledon who were finishing the half strongly and when the ball ricocheted kindly into Ethan Chistlett’s path, Eastwood had to come off his line to make a key save.
And they did have a second goal when Cheye Alexender got free down the right and found Assal, who squared for Pressley to slot home.
Pompey then perked up dramatically in stoppage-time and hit the woodwork twice in quick succession.
Harrison did well to find Hackett-Fairchild and he hit a rocket against the crossbar, while Hirst’s follow-up header struck the same part of the frame.
The Blues did halve their deficit seconds before the interval, though, as Thompson stole possession and Oualah made a mess of dealing with the shot from HARRISON.
AFC Wimbledon 2
Pompey had switched to a back three midway through the first half and continued with that once the action restarted.
And they immediately set about trying to find a leveller, with Oualah escaping any punishment for failing to hold a Hackett-Fairchild corner.
Another flag-kick from the wing-back led to the Blues’ leveller, however, as his centre was nodded back by Ogilvie at the back post.
HARRISON was surrounded by defenders, but showed plenty of strength to rise above them and firmly head home.
The hosts started to come back into the contest, although an Alexander header lacked the power to trouble Eastwood.
Thompson was starting to look a real presence in Pompey’s midfield, but – probably with his recent return to fitness in mind – was replaced by academy midfielder Harry Jewitt-White on 71 minutes.
Pompey kept pushing for a winner and almost went ahead for the first time when a Hackett-Fairchild corner was only cleared as far as Jacobs on the edge of the box.
His low drive was diverted towards the target by Hirst, but Oualah used his legs to prevent the striker from opening his account.
And they did go in front when Jacobs used the outside of his boot to spray the ball out to Hackett-Fairchild on the left.
He squared it for Gassan inside the box and although the loanee’s attempt was blocked, HARRISON pounced to convert the rebound.
The lead did not last long, however, as Mnoga was penalised for a foul on Henry Lawrence inside the area and Pressley converted from the spot.
Pompey then immediately came close to another goal of their own, with a Hirst header clawed away by Oualah.
But a topsy-turvy contest swung back in Wimbledon’s favour as Anthony Hartigan’s free-kick was nodded into the net by Nightingale.
And with the Blues committing most of their players forward in stoppage-time, the hosts broke up the pitch to score again.
Eastwood did well to deny Pressley his own hat-trick, but the ball ran loose for substitute McCormick to score.
AFC Wimbledon (4-2-3-1): Oualah; Alexander, Kalambayi (c), Nightingale, Guinness-Walker; Marsh, Hartigan; Assal (McCormick 70), Chislett (Rudoni 70), Mebude (Lawrence 57); Pressley
Goals: Kalambayi 2, Pressley 43, 80 (pen), Nightingale 87, McCormick 90+4
Subs not used: Tzanev, Heneghan, Woodyard, Bartley
Pompey (4-4-2): Eastwood; Romeo, Mnoga, Downing (c), Ogilvie; Hackett-Fairchild, Thompson (Jewitt-White 71), Jacobs, Gassan; Hirst, Harrison
Goals: Harrison 45+4, 52, 78
Subs not used: Steward, Kaba, Simpson, Bridgman, Payce, Gifford
Referee: Darren Drysdale