City shaded a tight and cagey first half, but neither side was able to make a breakthrough in the League One clash at St James Park.
David Noble's long-range shot was deflected over the Argyle crossbar, while Daniel Nardiello got on the end of a flowing Grecians move but could only poke the ball wide.
Argyle rarely threatened until just before half-time when Exeter goalkeeper Artur Krysiak was adjudged to have picked up a Richard Duffy backpass six yards from his own goal line. The ball was laid of to Luke Summerfield, but the Exeter defence held firm to keep out his thunderous shot.
City took the lead in the 46th minute when Dunne latched onto Jamie Cureton's cute flick and lashed a stunning volley past David Button from 18 yards.
Argyle, with nothing to lose, went for broke and Rory Fallon had the ball in the net only for his effort to be disallowed for offside, while Yannick Bolasie twice went close for the relegation-threatened Pilgrims as Exeter held on for the win.