After the match ended all square at the Merseyrail Community Stadium, the Rovers goalkeeper saved three penalties in the resulting shootout to earn a last four tie at home to Fleetwood Town.
In bitterly cold conditions, the home side edged the first half, Ben McKenna striking a post, whilst Fisher tipped over a Liam Nolan header.
But the goals arrived in the second half as Jamie Allen fired the Sandgrounders in front on 57 minutes, only for Lewis Mansell to brilliantly curl home an equaliser just three minutes later.
Rovers pressed hard for a winner but weren unable to grab that key second goal meaning the tie would be settled by spot-kicks.
The first five penalties were all successfully converted, but with Fisher saving the next three from Southport, it meant Rovers were the team to progress.
Damien Johnson made just one change from the side that defeated West Ham United on Friday night as Daniel Butterworth was brought in for only his second start for the Under-23s, Anthony Stokes the man he replaced.
In contrast to the young Rovers team, Southport named a side with an abundance of first team experience.
And on a sticky surface it was the hosts who has the first meaningful chance of the match on nine minutes. A corner from the right picked out Southport captain Jack Higgins, but Andy Fisher in the Rovers goal saved.
Rovers responded with an excellent chance of their own. A free-kick just inside the opposition half was sent long by Jack Doyle to Matty Platt at the back post, but he headed straight at keeper Magnus Norman and as it rebounded back towards the defender Platt was unable to force it home.
The early flurry of chances continued as Fisher denied Ryan Higgins, before following good work from Ashley Grimes, Ben McKenna shot from the corner of the area but saw his effort across goal clip the far post.
As the half wore on it was the National League side who started to gain the other upper hand. Andrai Jones looping a header over the bar after a spell of pressure.
But on 36 minutes Southport should have taken the lead. A ball over the top saw Jamie Allen through on goal, but whilst his effort beat the onrushing Fisher, it trickled just past the post.
The Sandgrounders continued to press up to half-time and only a fantastic finger-tip save from Fisher prevented the home side from going in a goal to the good. Liam Nolan with a looping header that seemed destined for the back of the net until the Rovers keeper's timely touch.
Willem Tomlinson was the first with a sight of goal as play resumed in the second half, the Rovers midfielder letting fly with a trademark effort from distance, but Norman was equal to his efforts.
However on 57 minutes it was the home side who grabbed the opening goal. Rovers cleared the initial set-play but as the ball was played back into the box Allen had time and space to fire home despite the best of efforts from Fisher.
Rovers almost hit back straight away when Tomlinson and Lewis Mansell combined well for Butterworth, but the youngster was just over the bar with his strike.
But just a minute later the visitors grabbed the goal their play merited. Mansell raced forward with options left and right, but he opted to go it alone and hit a lovely curling shot into the top corner to make it 1-1.
Boosted by the goal Rovers went in search of a winner. Joe Rankin-Costello saw a deflected strike saved, before good work from Doyle saw Mansell just fail to get on the end of his cross, the ball coming out to Lewis Hardcastle but the midfielder fired over.
Rovers continued to threaten as Mansell saw a low shot kept out before a Doyle effort deflected narrowly wide.
But despite the home side being pressed back for long spells, it was they who could have secured progress through in injury-time at the end of the game. McKenna's ball into the box picked out Ashley Grimes but the forward spurned the opportunity as he shot over.
That meant the tie would be settled by penalty kicks. Butterworth, Hardcastle and Mansell were all successful for Rovers before Fisher took centre-stage.
The young stopper denied Declan Weeks with a save to his right, however Tomlinson fired over for Rovers.
Fisher then saved again to thwart Ryan Higgins, but Lewis Travis missed the chance to win the match when he put his penalty wide of goal to leave it still at 3-2.
But Fisher spared the defender's blushes with his third consecutive stop, saving Rory McKeown's effort with his legs to set up a last four tie at home against Fleetwood Town.
SOUTHPORT: Norman, R. Higgins, McKeown, Nolan, Cundy, J. Higgins (c) (sub White 55 mins), McKenna, Grimes, Allen, Hynes (sub Weeks 46 mins), Jones (sub Cofie 66 mins).
Subs not used: Springthorpe (GK), Pritchard.
Goals: Allen 57 mins.
ROVERS: Fisher, Travis, Doyle, Tomlinson, Magloire, Platt (c), Thomson, Hardcastle, Mansell, Butterworth, Rankin-Costello.
Subs not used: Jackson, Powell (GK), Mols, Grayson, Hendry.
Goals: Mansell 60 mins.
Booked: Rankin-Costello 63 mins.