A second-half strike from substitute Richard Walker lifted promotion-chasing Shrewsbury to fourth place, within a point of the League Two top three.
Kevin Ellison hit the post for City while Ritchie Partridge spurned a golden chance from close range.
Walker then bundled home a close-range winner after Chris Humphrey's low cross somehow made it through a crowded box.
Chester midfielder James Harris was shown a straight red card late on for a challenge on Humphrey.
Struggling City are only seven points clear of the drop zone, having drawn just two games and lost eight (six of them now in a row!) in a dreadful 10-game run since Boxing Day.
And 23rd-placed Bournemouth, who have picked up 11 points out of a possible 15 from an unbeaten five-game run, now have two games in hand.
Bookmark with:
What are these?