But despite losing the golden-point toss, the Eels took advantage of a questionable penalty early in the first set of the third period when Michael Lichaa was penalised for stripping.
"I thought the penalty in the extra time was a pretty tough call, to be honest. I think he got it wrong. But we had other opportunities. It was a tough game," Bulldogs coach Des Hasler said.
The Eels then needed just three plays before Moses stepped up in the clutch to deliver his best moment in Parramatta colours since moving from the Wests Tigers mid-season.
It was easily Moses' best performance for the Eels, with him playing a hand in both their tries and coming up with some big defensive plays, including a try-saver on Josh Morris.
Lock Nathan Brown was named man of the match after producing a game-high 228m and 42 tackles in an 82-minute performance.
The signs were ominous early for Canterbury, who were on the backfoot in just the third minute when an errant first-tackle pass from Will Hopoate should have ended in a penalty try.
Instead, winger Marcelo Montoya was sin-binned for preventing a certain four-pointer from Mitchell Moses and Gutherson kicked an early penalty goal.
Down to 12 men, the Bulldogs twice bombed opportunities to take the lead and instead settled for their own two points gifted by Edwards.
However, that was as threatening as the visitors got as the home side took a half-time lead through tries to wing pair Bevan French and Semi Radradra.
Bulldogs captain Aiden Tolman's matched Brown's whole-game effort with 147m and a game-high 54 tackles in the defeat.
Eels coach Brad Arthur said there was no excuse for their second-half display.
"Maybe I need to rip into the boys a bit more just before we go out after half-time. They just had some big bodies out there, carried hard and aggressive," he said.
- AAP
