The visitors eventually opened the scoring five minutes before the main break, with Reynolds lobbing a kick into the path of Alex Johnston to touch down out wide.
Reynolds nailed the conversion from the sideline for a 6-0 lead, then Greg Inglis almost doubled the advantage two minutes later, with his angled run cutting through the Storm defence before the video review ruled obstruction.
Melbourne responded immediately after halftime, with Marika Koroibete diving over in the corner to level the scores.
The Storm appeared to click into gear, finally able to break through the Rabbitohs defensive line.
With Jesse Bromwich eating up metres through the middle, the ball went wide for Cheyse Blair to put the home side ahead 10-6.
But the Rabbitohs struck back after a Sam Burgess offload released Greg Inglis deep into Storm territory.
From the next set of six, Reynolds went to the air again for Angus Crichton to score his first NRL try and give his side a 12-10 advantage.
A Smith penalty goal levelled the scores at 12-all but a penalty against Christian Welch for a crusher tackle allowed Reynolds to regain the lead with 10 minutes remaining.
However, the Rabbitohs could not hold on and despite a much improved performance, slumped to their ninth straight defeat.
Storm coach Craig Bellamy praised his skipper for the late-game heroics.
"We got that penalty there and Smithy stepped up," said Bellamy. "That was a great kick under pressure.
"And to kick that field goal as well," added Bellamy. "That's why he is a special player.
"We were very lucky tonight and it was a couple of big plays from Cam that got us home."
While feeling his side were off the pace, particularly in the opening half, Bellamy was full of praise for the Rabbitohs.
"They had a tough week last week and we knew there would be some sort of bounce-back effect from that.
"I thought they defended really well tonight. It probably disappoints me to say it but I think they out-defended us tonight."
Despite the loss, Souths coach Michael Maguire was proud of his team's efforts, particularly in defence.
"The Storm threw plenty at us," said Maguire. "Our boys really dug deep many times to repel them and put ourselves in a great opportunity at the back end but it just wasn't to be.
"We have been working hard to achieve that type of performance for the past couple of months," added Maguire.
"We will get a lot out of that game. I think the boys really found what the South Sydney rugby league team is all about."
- AAP