It is indeed a stylistic inversion. It has the verb of motion (went) which is being used metaphorically, and it is inverted. It is exemplified on a forum post
The starter's gun went bang and the runners went off at a good pace.
Bang went the starter's gun and the runners went off at a good pace.
Although you could go further and say
Bang went the starter's gun, and off the runners went, at a good pace.
The purpose is to bring the short onomatopoeia "click" to head the phrase "click went her face", and give end weight to the longer "went her face"