A close contest, played in sweltering conditions, was decided by clinical finishes from Raheem Hanley and Deniz Pero.
Alan Wright's Conference side had their chances, John O'Sullivan cleared against his own post in the first half, whilst former Rover Osayamen Osawe had the best chance in the second with a header that struck the woodwork.
In the Southport sunshine it was the home side who threatened with the first chance after just three minutes. An inswinging corner caused consternation amongst the Rovers ranks, John O'Sullivan having to clear off his own goal line via the upright.
Rovers responded with a fifth minute effort, Jordan Slew firing narrowly wide following a quickly taken free kick.
The game continued to go from end to end, Southport's Danny Hattersley heading wide when well placed before goalkeeper Danny Hurst saved well after Jordan Slew's mazy run into the area.
Moments later Hurst was called up again, diving bravely at the feet of Curtis Haley to claim the ball in the area.
The Sandgrounders threatened once more with a corner from the right, captain Jamie Milligan's sending the ball into the box but Hattersley couldn't get a meaningful connection as the ball flashed wide of goal.
As the heat started to slow play down Ryan Edwards had the last chance of the first half, the defender letting fly from distance requiring Hurst to save once more, diving to his left to push the stinging effort around the post.
Into the second half and John O'Sullivan tested substitute goalkeeper David Stevenson on 51 minutes, the Irishman's low drive needing to be well saved.
As the half wore on both sides made a host of substitutions which disrupted the flow to the game somewhat, Matty Flynn heading narrowly over for Stockport before Liam Kearney saw his header well saved.
With Southport pushing for the opener, they were caught on the break on 70 minutes. Raheem Hanley, who impressed in his 45 minute outing last Saturday against Clitheroe, started and finished the move, racing from inside his own half to take on Livesey on the edge of the area before firing low into the far corner.
Three minutes later and Southport were within a whisker of equalising. Former Rover Osayamen Osawe was left unmarked at a corner but his header across the box struck the far post and bounced to safety.
The home side continued to press with several half chances, before David Fitzpatrick saw his shot from distance pushed aside.
A minute from time Rovers added a second goal, a cross from the right was mis-controlled by Matty Challoner, his marker Deniz Pero took full advantage and steered the ball into the bottom corner.
There was still a chance at the death for the Sandgrounders but Wayne Curtis just couldn't connect in the area.
Eric Kinder's side are back in action next Saturday at Harrogate Town.