Edwards, the man of the match, was quick to react when Sunderland goalkeeper Anthony Patterson made a fantastic save on Kornel Szucs' powerful shot in the dying moments of the match.
Edwards let fly with an equally venomous strike which gave the Sunderland number one no chance.
A Daniel Ballard own goal and Ryan Hardie's penalty had seen Plymouth come back from behind after Patrick Roberts' first-half penalty, with Romaine Mundle equalising late on for Sunderland before Edwards had the final say.
Sunderland went close in the opening minute as Mundle's pacy cross from the left flew across the face of goal and somehow eluded his team-mates.
TrendingJobe Bellingham fired high and wide from 20 yards for the visitors before Argyle keeper Dan Grimshaw - making his Championship debut in place of the injured Conor Hazard - saved Chris Rigg's angled strike.
But the Black Cats took a 24th-minute lead from Roberts' left-footed penalty, sending Grimshaw the wrong way after Bali Mumba, making his 100th Argyle appearance, was adjudged to have fouled Chris Rigg on a run into the box.
Referee John Brooks - well placed a few yards away - had no hesitation in pointing to the spot.
Argyle had a good chance to respond in the 32nd minute when the ball broke to Darko Gyabi, unmarked on the edge of the box, but the 20-year-old midfielder's first-time strike flew over.
Homegrown midfielder Adam Randell let fly with a thumping 25-yarder which Patterson was forced to go full length to tip around his post after 49 minutes.
It was the closest Argyle had come to scoring against their high-flying visitors.
The Greens restored parity after 54 minutes following a spell of sustained pressure.
Ibrahim Cissoko's cross from the right fell to Callum Wright, whose shot was blocked.
The ball fell to Morgan Whittaker, who hammered the ball past Patterson and in off the bar off defender Ballard.
Then when Wright's 61st-minute cross was cleared, it fell to Edwards, who let fly with a volley from the right which skimmed the far post.
New Swedish attacking midfielder Rami Al Hajj announced his arrival - as a 65th-minute substitute - by firing just wide.
Cissoko came close to putting Argyle ahead in the 68th minute when he cut in from the left and let fly. His shot beat Patterson but crashed off the crossbar.
The home side went ahead in the 73rd minute when substitute Hardie was fouled as he attempted to latch on to a cross from Cissoko. Brooks was on the spot to award a penalty, which was coolly converted by the Scottish striker.
Sunderland levelled in the 86th minute when Mundle's shot from the left went through a crowded penalty area and in at the far corner before Edwards' dramatic clincher.
Plymouth's Wayne Rooney:
To follow...
Sunderland's Regis Le Bris:
To follow...