ずに済む

「ずに済む」expresses the idea of "to get by without doing action V."
This structure is used when the speaker already knows in advance, has planned ahead, so the matter is resolved without needing to take action; or when encountering something that should be done but then managing to avoid it, not having to do it. Because these things are often troublesome,「ずに済む」reflects the speaker's light-hearted and relieved attitude.
It is often used in written language. In spoken language,「なくて済む/ないで済む」is used.