The visitors raced into a three-goal first-half lead at Hampden.
Weatherson opened the scoring with an unstoppable 15th-minute free-kick, and he made it 2-0 in the 27th minute.
Matt Flynn made it 3-0 six minutes later with a fine finish.
Queen's pulled a consolation goal back in stoppage time through John Carter.
Peter Weatherson netted a brace as Annan recorded only their second win in eight matches.