before one could; in no time at all; without giving a chance
Fixed adverbial phrase used after a plain verb to mean that something happens so quickly or overwhelmingly that the other party has no opening to react. Often used in competitive or confrontational contexts.
The opponent attacked without giving us any opening.
He answered the question before one could even react.
From 隙 (opening, gap) + も (even) + 与えず (negative continuative of 与える, 'to give'), literally 'without even giving a gap'. The expression is a fixed adverbial phrase.