Brough was yellow carded shortly before the half-time break and those 10 minutes proved vital as Catalans ran in 12 points to take the lead.
But Brough returned to the field to see his side to victory, coming from 12-6 down to lead 14-12 within five minutes of his return.
After three successive defeats away from home, the Dragons were looking to get back to winning ways at on their own field - where they had lost just once in 2015.
The Giants were also looking to get back on the right track after a narrow defeat to St Helens last weekend.
In an error-strewn game, the Giants just edged the two points after managing to keep a hold of the ball better than the hosts in hot and humid conditions.
Huddersfield took the lead in the 28th minute after an eight-minute period that saw the Dragons fail to touch the ball.
Jamie Ellis threw a pass wide to Leroy Cudjoe and the Giants centre drew in the cover defenders and offloaded to Jermaine McGillvary who scored in the right corner. Ellis added the conversion.
The Dragons levelled in the minute before half-time, with Brough in the sin-bin Catalans took advantage of the extra man. Krisnan Inu pounced on Morgan Escare’s neat kick to level the game at the break.
After half-time Inu turned provider as he broke clear before handing on to Escare who raced 70 metres unopposed to score. Scott Dureau added both conversions to take the score to 12-6 in favour of the home side.
But Brough returned to the field and had a hand an instant impact.
Aided by a penalty, the Giants attacked down the right and Scott Grix’s wide pass found McGillvary, who scored his second try of the afternoon.
Ellis converted, then added a penalty five minutes later to give the visitors a 14-12 lead after Dureau held on to Ukuma Ta’ai too long.
It was a lead Huddersfield would not surrender and they returned to Yorkshire with two points between them in fourth and Castleford in fifth.