Josh Todd's brace of strikes had Annan seemingly in control.
Lewis Guy appealed in vain for a penalty which could have finished the game before Sean Higgins headed one back for Clyde.
Strikes by Scott McLaughlin and Higgins turned the match on its head only for substitute Weatherson to curl in a superb 25-yard free-kick.
Peter Weatherson salvaged a draw for Annan Athletic after they had earlier let a two-goal lead slip against Clyde.