Young St George Illawarra centre Max Feagai was also forced off with an ankle injury in the second half at CommBank Stadium on Sunday evening.
The Dragons fought back from 16-0 to be level at half-time as English prop George Burgess got through his first match since late 2020 following radical hip-resurfacing surgery.
The former South Sydney bookend, who left the NRL after the 2019 season to join Super League club Wigan, finished with seven runs, 55 metres and five tackles in a solid stint off the bench.
Parramatta confirmed winger Dunster sustained "ligament damage" and will undergo scans after the winger was awkwardly caught underneath Fuimaono in the 13th minute.
The St George Illawarra second-rower was sin-binned and placed on report for a hip drop tackle.
Due to his rap sheet, which includes a grade-two hip-drop offence in 2020 and a reckless high tackle last season, Fuimaono would have loading and carryover points applied to a potential charge.
Just two minutes after the Indigenous All Star was given his marching orders, Moga, in his first appearance in the Red V, was sin-binned for clocking new Eels winger Bailey Simonsson high.
Matterson, a late addition to Parramatta's squad, joined them off the field after a high tackle.
Eels hooker Reed Mahoney returned from the shoulder injury that curtailed his 2021 campaign and his team were right on top while he was directing play around the ruck.
Parramatta had all the early ascendancy, catching out St George Illawarra's right edge with quick hands down the shortside as Hayze Perham scored in the third minute.
Front-rower Oregon Kaufusi barged over in the 11th minute before Sean Russell, replacing Dunster on the left flank, made it three unanswered tries.
After starting at fullback, Cody Ramsey traded places with Tyrell Sloan on the wing during the first half as Dragons coach Anthony Griffin cast his eye over his No.1 options.
Feagai, Connor Muhleisen and Moga - courtesy of a neat offload from centre recruit Moses Suli - grabbed four-pointers as St George Illawarra found momentum.
Solomone Naiduki put the Eels back in front in the 44th minute before Josh McGuire executed a run-around play to create space for Dragons winger Bailey Antrobus to score.
Sloan delivered a beautiful cut-out pass to give Jonathan Reuben the match-winner in the 63rd minute.
