The Vanarama Conference strugglers looked to have earned a replay but suffered heartbreak in the third minute of added time, when skipper Luke Foster brought down Johnny Russell and Chris Martin made no mistake from the spot.
The Sky Bet Championship outfit had struggled to find a breakthrough, as visiting keeper David Raya Martin made a string of magnificent saves.
Derby made nine changes from the team that beat Leeds midweek, but needed to introduce first team regulars Jordon Ibe, Chris Martin and Jonny Russell in the second half to try and force a result.
Despite being 89 places below Derby in the football pyramid, the 2,700 visiting Southport fans were given early encouragement when a weak header by Lee Naylor let in Mitchell Austin in the fifth minute, forcing Derby keeper Kelle Roos to dive at his feet to push the ball away.
Southport threatened again in the 11th minute when Paul Rutherford got behind Derby on the left, only for his cross to be blocked.
Derby should have scored in the 20th minute when Leon Best blazed wildly over from eight yards out, after Craig Bryson’s low cross-shot had come back off Raya Martin’s post.
The pressure was building for Southport and Raya Martin had to dive to his left to keep out a 25-yard drive from Paul Coutts in the 35th minute, before Simon Dawkins shot over from the edge of the area five minutes later.
The hosts wasted another great chance in the 42nd minute when Dawkins and Naylor combined down the left, only for Ryan Shotton to head wide from six yards.
Southport had been on the back foot for long periods but came close in the 58th minute, but Richard Brodie couldn’t get enough power on his header from a deflected cross.
The visitors came close again two minutes later when Kay had a shot blocked and it was a sign of Derby’s increasing anxiety when McClaren made three changes in the 63rd minute.
The Rams still could not break Southport down and the outstanding Raya Martin made another superb save in the 78th minute when he flew to his right to parry a Jake Buxton header.
However, Southport’s resistance was finally broken and although Raya Martin went the right way, Martin’s penalty was too powerful and Derby breathed a huge sigh of relief.