Graham Webster put the Gable Endies ahead in the 29th minute and defender Cammy Ballantyne added a second with 13 minutes left to leave the hosts bottom after a third straight defeat.
Falkirk are second on goal difference after they beat Clyde 3-0 with a brace from Callumn Morrison.
Paul Dixon gave the hosts the lead just before the hour, with Morrison adding another from the penalty spot in the 67th minute and knocking in a third during stoppage time.
Queen's Park are also unbeaten, but drop to third following a 1-1 draw at Alloa.
Simon Murray headed the visitors in front early in the second half, only for Steven Boyd's free-kick with 12 minutes left to secure Alloa a point.
Cove Rangers won 3-1 at Dumbarton.
Two goals in as many minutes midway through the first half from Mitch Megginson (23) and Robbie Leitch (24) put Cove in control.
Although Stuart Carswell reduced the deficit at the start of the second half, Rory McAllister wrapped things up when he slotted in a penalty with six minutes left.
Airdrieonians battled to their first league win with a 3-2 victory at Peterhead, who threatened a late comeback.
Rhys McCabe had given the visitors a half-time lead, and then added a second with a 70th-minute penalty before midfielder Gabriel McGill scored a third five minutes later.
Scott Brown pulled one back with nine minutes to go and substitute Russell McLean knocked in another in the 90th minute to set up a tense finale, but Airdrieonians held out.