Francois Zoko put the Glovers ahead - against the run of play - in first-half stoppage time.
Orient - with fit-again, Liam Kelly, impressive in midfield - had dominated for long spells.
Recalled, Rowan Liburd and Sandro Semedo failed to take advantage when well-placed.
Prolonged treatment to home keeper, Artur Krysiak, after a sickening clash with Liburd, proved costly.
His kick-out saw Eaves bring a splendid save from Sam Sargeant, but no Orient defenders reacted quick enough and Zoko gleefully netted with an angled drive.