Fevered expectation among the home fans after the early dismissal of Swansea forward Jordan Ayew gave way to frustration as the visitors repelled wave after wave of attacks, with Mooy - who lasted 69 minutes on his return from a knee injury - unable to inspire the home side to victory.
Swansea goalkeeper Lukasz Fabianski produced a world-class save to keep out Huddersfield striker Steve Mounie's second-half volley and Tom Ince's last-minute header bounced up against a post.
Swansea, who had collected 17 points from their previous nine league games, looked well-organised and resolute in the early stages.
But Huddersfield's task appeared to have been made a whole lot easier in the 11th minute when the Welsh side were reduced to 10 men.
Ayew lunged in studs up on Jonathan Hogg, catching the Terriers midfielder halfway up his shin and referee Michael Oliver did not take long to brandish a straight red card.
Town looked to quickly capitalise and Hogg's shot was blocked before Christopher Schindler's header sailed over.
Ince spurned a decent chance by dithering inside the penalty area and Mounie's improvised header from Mooy's raking cross flew narrowly over.
For all the home side's dominance, Ince's 45th-minute free-kick - easily held by Swansea goalkeeper Fabianski - was their first effort on target.
Hogg crashed a low 30-yard shot just wide soon after the restart as Huddersfield resumed their siege on the Swansea goal.
Town left-back Scott Malone wasted the game's best chance when he completely missed the excellent Florent Hadergjonaj's cross from in front of goal.
Mounie twice went close in quick succession, firing Hadergjonaj's cut-back into the side-netting before forcing Fabianski into a brilliant reaction save with a volley from inside the area.
Hadergjonaj curled a shot agonisingly wide direct from Mooy's low corner, Mounie directed another header off target and Malone's shot was blocked by Swans centre-half Mike van der Hoorn.
Both clubs were in English football's bottom tier in the 2003-2004 season and, on this evidence, have plenty of fight left in them as they bid to preserve their top-flight status.
Swansea, buoyed by last week's biggest win of the season against West Ham United, hardly stepped foot out of their own half, but refused to buckle.
Huddersfield, hoping to make their top-flight status as secure as possible before a daunting run-in, never took their foot off the pedal.
Swansea, beaten only once now in 14 matches, were content to let the home side have the ball and, save for a handful of clear chances, held on for the point they travelled north for.