Tedesco was ruled out of the match in the first half following a high hit from Bailey Simonsson, and Suaalii rose to the challenge to be among the Roosters' best in one of his first NRL games at the back.
The 19-year-old had a hand in the three tries for the Roosters, a week after signing a $4.8 million three-year deal with rugby union from 2025.
His performance helped the Roosters claim their biggest win of the season, their attack looking slick through a willingness to move the ball though the middle.
Tedesco's injury is likely to have longer-term ramifications with the star No.1 set to miss next Thursday's clash with Melbourne after Simonsson hit him high after a pass.
Joey Manu will return for that match from suspension with Roosters coach Trent Robinson having to make a call on whether to play Manu at No.1 or keep Suaalii there.
Canterbury supremo Phil Gould claimed this week every time Suaalii did anything from now on it was producing free advertising for rugby.
If that was the case the 15-man game had plenty to gain out of Allianz Stadium on Thursday night.
Playing at centre to start the game, Suaalii was instrumental in the Roosters' first when he put them on the attack by stepping past Mitch Moses to break into space.
From the ensuing plays, Sam Walker threw a long harbour-bridge ball for Jaxson Paulo, who put Drew Hutchison over with a flick pass.
With the Eels down to 12 men and with Simonsson off the field, Suaalii and Luke Keary helped Corey Allan put Daniel Tupou over.
Suaalii's third big involvement came when he swept around to the right and produced a flick pass for Jaxson Paulo to score, after seven straight sets on the Eels' line.
Keary also bagged a double for the Roosters with his first off the back of a Walker bust back through the middle and his second following a Tupou break.
Parramatta were gutsy after being reduced to 12 men twice in the first half, but are still with just one win from their opening five games.
Clint Gutherson and Moses both pulled off brilliant try-savers in the first half that typified their resolve to trail 16-2 at the break.
But they desperately missed Junior Paulo in the middle, while the Roosters weight of possession meant Reagan Campbell-Gillard spent most of his time defending.
Gutherson briefly put Parramatta back in the game at 16-8 early in the second half off the back of a Will Penisini flick pass.
But otherwise their tries came late with Simonsson and Maika Sivo both crossing in the final four minutes.
