Annan made the breakthrough 10 minutes from half-time after Josh Todd headed home Scott Davidson's cross.
It could have been 2-0 shortly afterwards when Mick Dunlop sent a header against his own crossbar.
Seven minutes after the break, Kenny Mackay doubled their lead with a fierce shot from just inside the area to make the three points safe.
Clyde slipped to a surprise 2-1 home defeat at the hands of bottom club Queen's Park, who are still without a permanent manager.
Blair Spittal hit the post early on for Queen's, but Stefan McCluskey put the home side ahead midway through the half.
Queen's hit back through a Spittal penalty and then Tom Collins struck in the second half to record only their second league win of the season.
Stirling Albion edged East Stirlingshire 2-1 at Forthbank when Mark Ferry raced clear in injury-time to hit an outstanding winning goal.
In an evenly-matched game, East Stirlingshire will feel aggrieved, having led until the final 10 minutes.
Max Wright gave them a half-time lead, but substitute Craig Comrie marked his debut by bagging the equaliser with his first touch of the ball with 10 minutes left. East Stirling had barely recovered from that blow when Ferry struck the winning goal.