Huddersfield dominated the first half, scoring three tries to go in at the break with an 18-0 lead, but - having been outclassed for 40 minutes - Tigers were in front at the hour mark.
Aided by two Giants sin-binnings, Tigers scored four tries to turn the game on its head, but the visitors’ second yellow card of the afternoon led to an equalising try and they were still a man down when Huddersfield regained the lead nine minutes from time.
The result was still in the balance at that stage, but Leutele’s third successive touchdown, with two minutes left, finally ended Tigers’ hopes.
It looked like being a routine Giants win in the opening period.
After a two-match ban, Danny Levi returned at hooker for the hosts and his work out of dummy-half caused Castleford all manner of problems, including laying on the opening try for Owen Trout who burrowed over from close-range after eight minutes.
Five minutes later, Chris McQueen made an excellent leaping catch from Oliver Russell’s cross kick and offloaded to send Luke Yates over.
Will Pryce converted both and landed a penalty 10 minutes before the interval after Jake Trueman was sin-binned for pulling back McQueen as he chased Tui Lolohea’s kick behind the line.
Referee James Child passed the decision on as no try and video assistant Jack Smith agreed.
When Jermaine McGillvary went over for Giants’ third try, from Leroy Cudjoe’s pass, it seemed the hosts had the game in the bag, despite Levi being shown a yellow card just before the interval for a late hit on Joe Westerman.
But, out of nowhere, Tigers made a rip-roaring start to the second half, scoring two tries in 11 minutes to cut the gap to just six points.
Westerman’s pass sent Liam Watts tearing clear and he turned the ball inside for Trueman to score.
Then Mahe Fonua ducked through from acting-half, Gareth O’Brien converted both and Tigers suddenly had the momentum.
Giants were back down to 12 men on 56 minutes when McGillvary was sin-binned for a late hit on O’Brien, whose pass in the set from the penalty sent Greg Eden in for Tigers’ third try.
O’Brien added a touchline goal and what had appeared to be a rout was a level ball game.
But not for long, moments later Eden went clear down the left wing again and turned the ball inside for Trueman to score his second.
The conversion made it 24-18, but three minutes later George Lawler was sin-binned for a dangerous tackle and following the penalty, Leutele dummied through and Pryce converted off the touchline to square the scores with 15 minutes left.
Just before Lawler returned, Lolohea and Russell shifted the ball to Leutele and he was in again with Pryce adding another touchline goal.
Then :Leutele went in again in the final moments, from Lolohea's final ball and Pryce's sixth goal completed the scoring.