Both teams struggled to create chances before Jonathan Hogg was dismissed for Huddersfield, but Cardiff were unable to take advantage as the wait goes on for their first victory of the season.
Huddersfield started the game on the front foot, but Cardiff almost took the lead as early as the fifth minute, with Ben Hamer needing to be brave to smother Nathaniel Mendez-Laing's effort, although the goalkeeper was soon substituted due to an injury sustained in the collision.
The home side hit back through Terence Kongolo - who struck an effort into the side-netting - but chances were few and far between during the remainder of the first half.
The Terriers had the better of the game heading into the break but despite a sustained period of possession, Neil Etheridge remained untroubled in the Cardiff goal.
However, Huddersfield responded by creating their best opportunity of the match as Steve Mounie saw a header at the back post superbly kept out by Etheridge.
The midfield battle had been hotly contested, but tensions eventually boiled over after an hour when Hogg was involved in an off-the-ball incident with Arter that saw the home side's player dismissed.
Being reduced to 10 men resulted in Huddersfield going onto the back foot, and Cardiff sensed their opportunity to get their first victory of the season on the board.
Bobby Reid failed to net with a header and Murphy fired off target, but Huddersfield were able to hold on to earn their first point of the campaign.
HUDDERSFIELD TOWN (4-3-3) Hamer; Hadergjonaj, Jorgensen, Schindler, Kongolo; Hogg, Billing Mooy; Van La Parra, Diahkaby (Mbenza 68'), Mounie (Depoitre 80')
CARDIFF CITY (4-5-1) Etheridge; Manga, Bamba, Morrison, Bennett; Ralls; Murphy, Arter, Camarasa (Reid 75'), Mendez-Laing (Paterson 10'); Zohore (Ward 59')