Following a goalless first half, James Bolton was sent off after the break for bringing down Damian Reeves.
But, despite having a man less, the hosts took the lead when Hughes found the net from inside the box.
The win means Halifax have moved up to 19th in the table, two points ahead of 21st-placed Alty.
The visitors went close to taking the lead when Halifax defender Jake Hibbs cleared a Josh Ginnelly chance off the line.
Alty also had a chance to equalise late on but Reeves fired wide from inside the box.
A second-half strike from Connor Hughes saw 10-man Halifax leapfrog Altrincham and move out of the National League relegation zone.