なくて済む/ないで済む => get by without doing ~ N2
的 => about (change noun into na-adjective or adverb) N3
を[経「へ」]て => After A N1
Time, Situation
~をたて(~を経て)      => through; after; via N1
ずに済む => get by without doing ~ N2
て済む => sufficient by; no problem to; resolve by ~ N3
~ひょうしに(~拍子に)   => the moment [A], unintentionally / inadvertently caused something to happen N1