Johnathan Thurston and Robert Lui combined well in the halves for the visitors in the second half after they trailed by four points at the break.
A Lui try early in the second half put the Cowboys ahead before Josh Mansour regained the ascendancy for the Panthers courtesy of a great long ball from Soward.
A flying Antonio Winterstein tied things up at 22-all in the 68th minute.
Thurston missed the sideline conversion before Moylan struck true.
With a strong wind at their backs the visitors opened the scoring in just the third minute when Thurston put Kane Linnett over to lead 6-0 with the conversion.
But from their the home side dominated the majority of the first half with a favourable penalty count and Cowboys errors helping them.
So too did the Panthers aptitude, led by Soward.
Some clever passing on the Panthers right hand side with Dallin Watene-Zeleniak created the Panthers' halfback's first try in the 11th minute.
A Soward penalty goal in the 19th minute tied the scores up at 6-all before the former Dragon scored his second try in a great individual effort when he took on the Cowboys goal line defence in the 27th minute.
When Jamal Idris tapped back a Soward kick for Jeremy Latimore to score a converted try the Panthers led 16-6, six minutes out from the break.
A Glenn Hall try from a Gavin Cooper offload just before halftime pegged the Panthers lead back to 16-12.
Panthers coach Ivan Cleary praised the composure of Moylan.
"He likes the big moment, it doesn't phase him," he said.
"The bigger the moment the calmer he is.
"It was tough at the end there, an arm wrestle and we came out at the end of it, that is really pleasing."
Cowboys coach Paul Green said he couldn't understand the sinbinning of Matthew Wright in the 49th minute after Soward had made a long break and was brought down by Michael Morgan.
It was a crucial ruling with Mansour crossing soon after.
"I'm a bit perplexed ... the referee didn't know what to call," he said.
"What was the professional foul?
"I just think he didn't know what was going on."
Green, however, conceded his team had opportunities to claim the win.
"We just weren't composed enough to maintain pressure," he said.
"We fell into the trap of wanting to score on every play, trying to score a try there and then.
"We were in a position to win the game, we just weren't good enough in the end."
- AAP