"侥幸的脱险" can be translated into English as "narrow escape" or "lucky escape".
"Narrow escape" emphasizes that the escape was achieved by a very small margin or with great difficulty, implying a close call with danger.
"Lucky escape" highlights the element of luck involved in avoiding a dangerous or unpleasant situation.
Both expressions are commonly used in English to convey the idea of escaping danger by chance or with a small amount of fortune.