ずに済（す）む is a grammar construction used in Japanese to indicate that one can ‘get away without doing (A)’, or ‘it’s no problem without (A)’. This structure is just a mix of ず, a literary auxiliary verb that attaches to verbs in the same way as ない (and has the same meaning), に, and 済（す）む, a う-Verb meaning ‘to come to pass’, or ‘to end up’. Literally, it can be thought of as meaning ‘it will work out without (A)’.
As mentioned, ず connects to verbs in the same way as ない, so this structure will be seen after the verbs that they are negating.

• 今（いま）宿題（しゅくだい）をやっておけば後（あと）でやらずに済（す）むから、今（いま）のうちにやっておこう。
  If I do my homework now, I can get by without doing it later, so I shall do it right now.
• このVIPパスを使（つか）えば待（ま）たずに済（す）みます。
  If you have this VIP pass, you can get by without waiting.

As the primary element in this grammar point that controls the meaning is 済（す）む, other negations such as ないですむ, or なしですむ may also be seen. 無（な）しで functions as a suffix, so may be used directly after nouns, unlike ないで and ずに.

• 今月（こんげつ）契約（けいやく）すると初期（しょき）費用（ひよう）を支払（しはら）わないで済（す）むので、今月（こんげつ）中（ちゅう）に契約（けいやく）することをお勧（すす）めします。
  We recommend that you sign the contract within this month so that you can get by without paying the initial cost.
• 今年（ことし）の冬（ふゆ）は去年（きょねん）より暖（あたた）かかったので、ヒーターなしで済（す）みました。
  Since winter this year was warmer than last year, I got by without using the heater.

In most cases, (A) will be something that is thought of as unpleasant or bothersome, therefore highlighting the speaker's relief that it can be avoided.