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She somehow got away without mentioning her plan.
Literal
She [topic-は] somehow own [genitive-の] plan [genitive-の] thing [object-を] not-saying [without-ずに] managed.
ずに済ませる ('manage without doing X') is the active/causative cousin of ずに済む ('end up without doing'). Built on the classical negative ず + に, it means 'to settle / get through / manage without having to do X.' The ずに form is more literary than the everyday ないで; both work, but ずに carries a slightly elevated register. なんとか ('somehow, by some means') often pairs with such patterns to underscore that the avoidance took some doing — implying she may have skirted close to revealing it.