Craig Mackail-Smith scored late on for Peterborough but Ferguson's men were unable to force the draw.
Paul Sturrock had come under fire from Plymouth fans following the poor start to the season and needed this win at London Road, which ended a five-month wait for victory.
Ferguson handed a debut to Tottenham winger Danny Rose but it was the visitors who started brightly.
Argyle last won in April but were presented with a great chance to open the scoring in the fifth minute.
Charlie Lee, back in the Posh line-up after an ankle injury, fouled Fallon on the right flank and when James Paterson swung over the free-kick Mackie got ahead of his marker but headed wide from six yards out.
Peterborough forward Mackail-Smith then forced Romain Larrieu into a smart save with a powerful volley before heading tamely at Larrieu when well placed.
Rose then let fly from 25 yards as Ferguson's side stepped up the pace, but Mackail-Smith should have broken the deadlock in stoppage time at the end of the first half.
Rose and Russell Martin combined to play the ball into the near post, but Mackail-Smith saw his low effort well tipped around the post by the diving Larrieu.
Fallon headed straight at Joe Lewis at the start of the second period, but the visitors did move ahead in the 63rd minute.
Posh left-back Jamie Day was forced off through injury and as the defence was reorganising, Mackie found space inside the area to plant a shot across Lewis and into the corner.
The goal shook the hosts and Sturrock's side doubled their lead in the 69th minute.
A fine counter-attack ended with the ball at the feet of left-back Gary Sawyer and he picked out the arriving Fallon perfectly, with the former Swansea man heading powerfully past Lewis from close range.
Mackail-Smith dragged another effort wide as Posh searched for a way back and the persistent forward did reply with four minutes remaining.
Strike partner Aaron McLean wriggled free on the left of the penalty area and dinked in a cross which Mackail-Smith powerfully headed into the far corner.
But Sturrock's men held on for a morale-boosting win which lifts Argyle off the bottom.