Second-half goals from Ally Love and David Hopkirk had given the visitors a cushion and though Andrew Little halved the deficit with half an hour left, the equaliser remained elusive.
Chief executive Charles Green had blasted this Gers side as the worst in the club's history but rather than deliver their riposte on the pitch, they slumped to a first home defeat of the season at the hands of a side with just half their points haul.
It had taken them 20 minutes to test the Annan goalkeeper but though Anestis Argyriou's 20-yard had ample power, it was at comfortable height for Alex Mitchell to hold.
Little skipped past Iain Chisholm, only to drive wide soon after, but Hopkirk had unsettled the home fans with a string of surging runs and Dean Shiels' premature departure through injury on the half-hour further raised the tension.
Murray wasted a good chance just before the restart and then saw a shot deflected narrowly wide soon after, home jeers and Ally McCoist's half-time words failing to add impetus.
It came as no surprise that Love nudged in a Danny Orsi centre (48) moments later and when Murray sent a deep cross to the far post, Hopkirk bravely dived to inflict further damage (55).
Little's timely header (59) - his 22nd goal of the season - provided hope of a turnaround but Lee McCulloch squandered a fine chance to keep the fightback on course when he struck straight into Mitchell's shins.
A Mitchell palm denied the same player from 18 yards soon after and Annan had done enough, the Galabank side's victory deserved.