Cooke, the architect of so much good work this season, must now wait to see if his suspension for attempting to head-butt Brad Drew will keep him out of the Challenge Cup semi-final against St Helens in two weeks' time.
"I don't think he'll get suspended," said Hull's coach John Kear. "It was a scrappy, frustrating game and that might have led, in part, to the incident."
Hull put their visitors under pressure from the start, with the only surprise being that it took them six minutes to break through. Richard Swain was back after almost four months, and the quality of his passing from dummy-half opened up the left flank for Chris Chester to send Richie Barnett over.
Ten minutes later the winger was in again. This time it was Cooke who spread the ball, but it was Kirk Yeaman's superb one-handed pass that made the try possible. Cooke added his second difficult conversion.
Huddersfield squandered all their promising positions, but they got it right after 24 minutes when Chris Thorman put Paul Smith through.
The Giants gained another two points and a one-man advantage when Motu Tony was sent to the sin-bin for holding on in the tackle and Thorman landed a penalty to add to his conversion.
Some calmer handling of the ball could have seen Huddersfield make further progress, but Hull kept them out. As soon as Hull were back up to strength, however, Huddersfield struck, the creative Thorman sending James Evans away and Lee Greenwood supporting for a try on his debut to put his new club ahead at the break.
It was not a lead that lasted long, Paul King's sweetly timed pass sending Richard Horne through. The Giants then seized the advantage again when Paul March kicked and Chris Nero latched on to the loose ball.
The lead changed hands for the last time when Horne and Cooke set up Yeaman and then Stephen Kearney offloaded for Richard Whiting to score. Cooke's pass and Yeaman's strength brought him his second on the hour before Cooke departed with 14 minutes to play.
Huddersfield could have narrowed the gap if Stuart Donlan's try had not been disallowed for a forward pass. Thorman's pass to Eorl Crabtree also looked forward, but the try stood and enabled Huddersfield to finish on an upbeat note.
Hull: Tony; Saxton, Whiting, Yeaman, Barnett; Cooke, R Horne; Dowes, Swain, Thackray, McMenemy, Kearney, Chester. Substitutes used: King, Carvell, Brough, G Horne.
Huddersfield: St Hilaire; Greenwood, Evans, Clayton, Donlan; Thorman, March; McDonald, Drew, Jackson, Nero, Smith, O'Hare. Substitutes used: Gannon, Crabtree, Grayshon, White.
Referee: R Silverwood (Mirfield).Reuse content