Fallon charged down Wes Morgan's clearance after Paul Gallagher had caused panic in the Forest box, and saw the ball bounce off his shins and past goalkeeper Paul Smith.
The victory completed a total turnaround for Argyle, who two weeks earlier had slumped to the very bottom of the table after a second successive home defeat, with just two points from their first five fixtures.
Since then, victories on the road at Watford and Crystal Palace have transformed the Pilgrims' season.
Forest will count themselves unlucky not to grab a share of the spoils, with veteran striker Andy Cole twice being thwarted by goalkeeper Romain Larrieu at the start of the first half and seeing another shot roll agonizingly wide of the goalpost.
Chris Cohen was also denied by the Pilgrims' French goalkeeper, making his 200th league start for his only English club, while a late shout for a penalty to the visitors was also turned down.
Argyle will point to a penalty shout of their own, when Julian Bennett appeared to impede Jamie Mackie as the Argyle midfielder jostled to get on the end of Chris Clark's cross.
Mackie also hit the crossbar during a second half in which Argyle spent a majority of the time defending.
The game ended in uproar as Morgan was dismissed deep into injury-time for a second yellow card from referee Kevin Wright after fouling Yoann Folly.
Forest players were incensed at the harshness of the decision, with Wright blowing for full-time as soon as Argyle took the subsequent free-kick. Cole and Nathan Tyson were booked.