Beside Dispel magic, there are also spells that can remove curses. These abilities work on all curses, no exceptions.
Lesser Remove Curse (Mage) Removes 1 curse from a friendly target. There isn’t anything lesser about it, it works on all curses.
Remove Curse (Druid) Removes 1 curse from a friendly target.
Lastly, the Curse of Exhaustion is considered a snare effect where certain abilities are concerned. Paladins can remove it by using Blessing of Freedom. Mages can break it by using Blink. Gnomes can remove it by using their Escape Artist ability.