〜ずに済む・〜ないで済む
get away without, not have to
Verb ない-form minus ない + ずに済む · ない-form + で済む
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ずに済む means you managed to avoid something you expected to have to do, usually with relief: 友達が車で送ってくれたので、タクシーを使わずに済んだ, "a friend gave me a lift, so I didn't have to take a taxi". 済む means "to be settled, to end".
It's built like ずに (N4): 使わない → 使わずに済む. ないで済む means the same and is more common in speech.
The first half usually explains what saved you: booking early, having insurance, someone's help. It often appears in the past, 済んだ.
A related phrase, で済む after a noun, means "it ends with just that": 謝るだけで済む問題じゃない, "it's not a problem an apology will fix".
Examples
友達が車で送ってくれたので、タクシーを使わずに済んだ。
A friend gave me a lift, so I didn't have to take a taxi.
早めに予約したので、並ばずに済んだ。
I booked early, so I didn't have to queue.
薬を飲めば、手術を受けずに済む。
If you take the medicine, you won't need surgery.
保険に入っていたので、お金を払わずに済んだ。
I was insured, so I didn't have to pay anything.
謝ってくれたので、大きな問題にならずに済んだ。
They apologised, so it didn't turn into a big problem.