The night started poorly for Parramatta, with talismanic fullback and captain Jarryd Hayne pulling out after not recovering from a knee injury.
Hayne's withdrawal forced a backline reshuffle and the Dragons scored converted tries to forwards Mitchell Rein and Tyson Frizell in the first 15 minutes.
Moimoi, who had scored just one try in 21 previous games this season and averaged one every nine games through his career, put the voice back into the Parramatta supporters, who were deathly quiet after a disastrous start.
He claimed and grounded a high kick from halfback Luke Kelly for one of the more improbable tries of his career.
Parramatta were level ten minutes later after Joseph Paulo converted a try to Radradra, who strolled over from close range after a breakdown in the Dragons defence.
Moimoi powered his way over in the 37th minute to break a 12-12 deadlock.
The Dragons were level inside two minutes of the second half, with winger Daniel Vidot set up by a quick pass from Chase Stanley.
Fijian under-20 rugby representative Radradra, playing in his sixth first-grade game, reclaimed the lead for Parramatta when he sprinted just over 50 metres after being set up by five-eighth Ben Roberts.
Ken Sio, who switched from wing to fullback following Hayne's withdrawal, crossed on the hour to give the Eels a ten-point buffer.
Parramatta coach Ricky Stuart was happy with the way his team performed, especially after the embarrassment of last week's 60-point hiding by Melbourne.
"It's been a very very tough year, it's hard going, but that puts a little bit of joy in your life,'' Stuart said.
"I'll sleep a little bit easier tonight.''
Prop Tim Mannah, who captained the Eels in Hayne's absence, celebrated the Eels ability to hang in the game, something they haven't always done this year.
"Sometimes we've been a little guilty of throwing in the white towel a bit when we get off to a bad start, but the boys dug their heels in and worked hard for each other,'' Mannah said.
"Ricky spoke a lot before the game about pride. The boys spoke about being selfless and working for the guy next to you and I think all those little things added up and we came away with the result.''
Dragons coach Steve Price was disappointed his team was caught out on the short side after focusing on that area in the lead up.
"We're in a tough spot at the moment, we're in a bit of a flat spot,'' Price said.
"We've lost our way to winning at the moment. It is hard to take.''
- AAP
