It turns out ZenDesk uses more of an "OR" search, while most search engines use "AND" searches. This means Google ensures nearly all searched words are present in results, while ZenDesk piles on as many results as it can find with at least 2/3 of your words included.
While ZenDesk provides several useful search tips, I had a problem: even after training my support team on advanced search operators for half a year, they avoided searching in ZenDesk unless absolutely necessary.
Here are two scripts which narrow search results to require all searched words, and automatically sort by the most recent ticket update. They're not perfect, but our team is finally using the search tool again and quickly finding what they need.