。
She used quick wit to deflect the awkward question.
Literal
She [topic-は] quick-wit [object-を] greatly used [and-て] dicey question [object-を] dodged.
機転をきかせる ('to use one's wits') pairs 機転 ('quick thinking, wit') with きかす — the causative-style of きく ('to be effective'), giving 'to make one's wit work.' (The 利かせる kanji form is also seen.) Pair it with 大いに ('greatly, considerably'), and the sentence reads as someone deploying their wit in full force. かわす ('to dodge, to parry') is borrowed metaphorically from physical movement — its core sense is sidestepping a thrown object — into the verbal sphere of evading a tough question. きわどい ('risky, ticklish, on the edge') describes the kind of question whose answer could backfire.