Matthew Dennis put an end to Carlisle's scoring drought by scoring the only goal in the match against Barrow in the Cumbrian derby.
This much-needed win brought Carlisle within two points of safety in Sky Bet League Two.
Dennis scored the opening goal for Carlisle with their first shot, converting Cedwyn Scott's backheel in the 19th minute.
Kian Spence missed the target with a shot, while Acquah headed off target before the break.
Elliot Embleton's attempt went over the crossbar for Carlisle after the break.
In the final 20 minutes, Ben Jackson's strong effort was saved by Carlisle's goalkeeper Gabriel Breeze, who prevented Tyler Smith and Acquah from scoring as Carlisle held on for the important victory.