David Wagner's men have not scored in their past five home league games but had the better chances in the first half of this one, Steve Mounie unable to convert a pair of quality Chris Lowe crosses.
Zaha, who has scored eight times in his past 10 Premier League matches for Palace, made the hosts pay with a goal out of the top drawer shortly before half-time.
Aaron Mooy hit the post after the interval, but the Terriers remain just above the relegation zone after a third league loss of the season, while Palace moves into mid-table.
Chances were always likely to be scarce, but Mounie should have scored in the 17th minute, the striker heading over an excellent left-wing Lowe cross from eight yards out.
Another superb Lowe ball, this time from the right, only needed a touch from Mounie to break the deadlock, before an ambitious Christopher Schindler bicycle kick bounced wide.
Mounie's wastefulness was punished by a sensational solo goal from Zaha in the 38th minute.
Fed by Jordan Ayew 40 yards out on the left, Zaha carried the ball through a swarm of Huddersfield defenders, jinked into the box and whipped a superb shot across Jonas Lossl.
Mooy was unfortunate not to level when his 59th-minute strike hit the woodwork with Wayne Hennessey beaten, while a second for Palace was ruled out for a foul by James McArthur on Lossl, and another loss without a goal deepens Wagner's woes.
Struggling Huddersfield heads to Leicester City next weekend, while Palace will look to collect a first home point of the season against Newcastle United.