Short answer
Yes, a small amount of rosemary is safe for puppies. Rosemary in the quantities used for cooking is not a concern, and its extract is even used as a preservative in some commercial puppy foods. Just ensure it's a tiny sprinkle of the dried herb or a small fresh leaf, thoroughly mixed into their food. As with any new ingredient, introduce it slowly and watch for any signs of digestive sensitivity. If all seems well, it can be an occasional flavour addition.
Rosemary — Food safety facts
Safe to share · Dogs · full breakdown on the linked food page
Safe in food amounts. Rosemary extract is a common pet food preservative.
The full verdict, portion guidance and serving notes are collected on the Rosemary food facts page.