The Shire belied their status as basement battlers and kept Jim Chapman's men at bay in the first half.
Ryan McStay cleared Ross Fisher's header off the line, while Thomas Orr got on the end of a pinpoint cross, but headed just over the bar.
Despite Shire's bright start to the second half, Annan led when Todd flicked home Ben Jago's long shot.
Annan Athletic had to work hard for a gritty 1-0 win at East Stirlingshire with Josh Todd bagging the only goal.