The negativity looked like coming back to haunt the Dragons when reigning Dally M medallist Jarryd Hayne ignited the 18,293 pro-Eels crowd with a deft cut-out for Luke Burt to score out wide.
Burt's second sideline conversion of the game cut the margin to four with 17 minutes remaining, but Parramatta's momentum was destroyed minutes later when a thumping hit by Jarrod Saffy and Dan Hunt knocked the ball loose from Nathan Cayless's grasp from the kick-off.
The Dragons let that opportunity pass but they were again given a leg-up by the Eels five minutes from time when Fuifui Moimoi was penalised for a strip on Michael Weyman.
That enabled Soward to slot his third penalty of the night from point blank range.
Both sides played the best part of half a game down one player.
The Eels' Shane Shackleton suffered a hamstring tear and Dragons hooker Nathan Fien - who started the day celebrating the birth of his fourth child - was carried off with a suspected broken ankle.
The Dragons were deserved 12-6 leaders at the break, though they would have gone to the sheds ruing the fact that they hadn't been able to turn their positional dominance into a greater advantage.
Prop-cum-centre Matt Prior was twice stopped only centimetres from the line in the opening minutes before Fien grubbered for Ben Hornby to score just inside the dead ball line.
Hayne reminded everyone he was human by kicking out on the full from inside his own half.
The Dragons made him pay for the lapse as Jeremy Smith bounced out of a pair of feeble tackles to take the advantage to 12-0.
The Eels had played most of the game in their own half and it was from there they eventually got on the board when unlikely protagonist Jeff Robson dummied through and found Hayne in support before Eric Grothe went over untouched on the next play.
Dragons coach Wayne Bennett confirmed Fien's ankle was broken and praised his side's ability to adapt to its high casualty rate.
"We lost Nathan Fien which is a blow for us, Ben Hornby was in doubt right up to the match and we've got three centres out injured," Bennett said.
Eels coach Daniel Anderson said the difference between the two sides was Soward's brilliant kicking game and the Dragons' 94 per cent completion rate.
"They were very clinical and disciplined and buried us in field position," he said.
In Brisbane, the Broncos had to battle to hold out a resurgent Cowboys team.
Brisbane was ahead 20-0 at halftime, but the Cowboys came back in the second half, scoring three tries within 10 minutes to have the scores level at 24-all in the 70th minute.
The Broncos broke the deadlock five minutes later, when Denan Kemp made a 60m run to score in the corner.
- AAP
