The pint-sized striker hit the winner 15 minutes from time, his seventh goal in his last four matches, to earn Chester's fourth successive win and further ease their relegation worries.
The home side had started the brighter of the two teams and Gregg Blundell saw a good opportunity blocked and Dave Artell headed wide inside the first five minutes.
A good break on 11 minutes saw Matt Derbyshire only inches away from getting on the edge of Simon Spender's cross which flew across the face of goal.
A good chance came Chester's way on 25 minutes when Blundell had options to play in either Asamoah or Jake Edwards, but opted for the tall striker who sliced his opportunity wide from 12 yards out.
Five minutes before the interval Ben Davies played in Asamoah and the in-form striker saw his goal-bound effort blocked by a last-ditch challenge from Dennis Lawrence.
A dubious penalty award gave Chester the lead ten minutes after the interval when Danny Williams was adjudged to have tripped Blundell as the striker was about to pull the trigger.
Midfielder Davies showed nerves of steel to rifle the spot-kick home despite the best attempts of the away fans gathered behind that goal to put him off.
Ironically, the goal seemed to spur the visitors on and they drew level on 69 minutes when a left-wing cross from Darren Ferguson was nodded home by Lee McEvilly from close range.
With the game in the balance it took a mistake from Wrexham midfielder Matt Crowell to decisively swing the game Chester's way.
A careless pass on 75 minutes gave the ball to Stewart Drummond and his inch-perfect pass played in Asamoah who slid the ball past the advancing keeper for the winner.