Fullback Papenhuyzen sealed the Storm's 25-24 win with his second attempt at field goal in the 84th minute, booting the ball over from 20 metres out at Accor Stadium on Saturday night.
Latrell Mitchell had done his utmost to seal a Souths victory on return from State of Origin duty, including kicking to Tyrone Munro for the try that equalised the game with only four minutes to play.
But he missed a potentially game-winning sideline conversion thereafter before his only field goal attempt sailed wide from 45 metres out.
An error-prone Storm moved to extra time, sealing victory and jumping the Warriors into third spot on the ladder.
Mitchell threw cut-out passes down the left side for Johnston's 203rd, 204th and 205th career tries.
Johnston then intercepted a Jahrome Hughes pass for his fourth try of the night, moving only six behind the benchmark of 212 that Ken Irvine set in 1973.
The winger's hopes of reaching the record appeared in doubt when the 30-year-old went down with a serious Achilles injury late last season.
But he has since scored 11 tries in seven games this year, and with nine more to play in the regular season, is well on track to reach 213 by finals.
He also became the first player to score four or more tries in a game against Melbourne since the perennially-successful Storm entered the competition in 1998.
The loss is Wayne Bennett's 15th in a row against rival coach Craig Bellamy and came in the pair's record-breaking 44th time coaching against one another.
Bennett has not beaten his former protege in any match since 2016, with Souths now in the midst of a four-game losing streak that ensures they will finish the weekend second-last on the ladder.
The Storm scored in back-to-back sets to lead 12-0 after six minutes but Bellamy would have been furious that the Storm invited Souths back in with errors.
The attacking powerhouse managed only a 68 per cent completion rate in the first half.
Melbourne looked ready to break their hosts' hearts as Sua Fa'alogo fielded a Mitchell bomb on the 20-metre line to put Tyran Wishart in.
The Rabbitohs equalised when Mitchell dared to kick a long grubber through the line for Munro from 30 metres out on the third tackle.
But it was Papenhuyzen who had the last laugh.
