Cudjoe was among the Super League's Top 10 try scorers this season with 22
Crusaders' play-off dream was cruelly shattered by Leroy Cudjoe's dramatic, late match-winning try at the Galpharm.
Brian Noble's men were set for an upset when they led 10-6 with just 10 minutes left, thanks to a Rhys Hanbury try and three Clinton Schifcofske kicks.
But Huddersfield took advantage of Jordan James' sin-binning to score through Larne Patrick and lead 12-10.
And although Schifcofske's boot again levelled, Cudjoe went under the posts and Brett Hodgson added his third kick.
Having trailed for most of the match, the Giants were heading for sudden-death extra time when hooker Danny Brough picked up a stray Crusaders pass and sent Cudjoe on a 50-metre sprint for the winning try.
Hodgson had also earlier converted scrum-half Luke Robinson's superb early second-half try after Crusaders had gone in 8-0 up at the interval.
Huddersfield's Super League elimination play-off victory takes them through to next weekend's semi-finals, but Crusaders are out - and now have to face up to how they fare next season if, as has been reported, coach Noble moves on.
It was Crusaders' third defeat against the Giants this season.
They lost 30-12 on their last visit in July, shipping six tries, and were also well beaten 38-10, conceding seven tries at the Racecourse Ground in April.
But this one was a lot tighter - and only indiscipline, in going down to 12 men for the final 10 minutes cost them victory.
Huddersfield, who had lost all three of their previous play-off ties, deserved their victory, having scored three tries to one. But Nathan Brown's men will need to improve next week if they are to keep alive hopes of a first Grand Final appearance.
If anything, Crusaders earned most of the plaudits for a gutsy show which typified their efforts in reaching the play-offs under Noble, just 12 months on from collecting the wooden spoon.
Stand-off Jarrod Sammut was a constant threat, while the back three of Schifcofske, Luke Dyer and Nick Youngquest were brilliant in what was, at times, a desperate rearguard defensive effort.
Apart from an early break from Castleford-bound winger Youngquest, the visitors barely troubled the second-best defence in the League in the first half, with Giants captain Brett Hodgson as solid as a rock at the back on his final home appearance.
The only try of the first half came against the run of play.
Scrum-half Hanbury intercepted a Brough pass meant for Cudjoe five minutes before the break and went on an unopposed 45-metre run to the line.
Schifcofske landed the conversion, much to the delight of the Crusaders' colourful 1,000-strong support among the 5,869 crowd. And they then made it 8-0 just before half-time with a penalty after Huddersfield were caught offside from winger David Hodgson's knock-on.
The home side began their fightback within four minutes of the re-start when man of the match Robinson burst through the first line of defence and had the pace to finish off his own break from 40 metres out.
Hodgson's goal cut the deficit to two points and the Giants dominated the third quarter, only to lack the killer touch.
Schifcofske kept the visitors in front with a second penalty on 62 minutes but their line was under siege for long periods and only a timely interception by Sammut prevented David Hodgson scoring at the corner.
The game then took a decisive turn on 69 minutes when Crusaders substitute James was given a yellow card by referee Ben Thaler for preventing the Giants from taking a quick tap.
Within three minutes Huddersfield were level when prop Darrell Griffin charged up the middle and offloaded for Patrick to score his side's second try.
Brett Hodgson's conversion put the Giants in front for the first time but it was quickly all square when Patkick barged into Vince Mellars from the re-start and Schifcofske kicked his fourth goal.
As the tension mounted, Brough lined up a drop-goal attempt only to fumble the ball. And Crusaders were mounting one last bid for victory, in time added on, when Brough picked up a loose ball on halfway to send in Cudjoe over.
While speculation mounts about his own future, Noble confirmed that Youngquest, Rocky Trimarchi and Tommy Lee had made their final appearances for the club, while 18th man Jamie Thackray will also be leaving.
Huddersfield coach Nathan Brown said:
"The great thing is that the burden's gone. We've actually won a finals game and we don't have to worry about that any more.
"We knew it was going to be a hard game and it was.
"Defensively, we did a terrific job. The only try they got was from an intercept. But Robbo's commitment and effort was terrific and I thought Darrell Griffin was a stand-out today, he was sensational."
Crusaders coach Brian Noble said:
"We probably didn't get what we deserved. Towards the back end a couple of things went against us.
"I thought we were the better team and, but for a couple of lapses in concentration, we could have been in the mix for next week.
"There is no doubt the sin binning had an effect because it gave Huddersfield a bit of momentum. It changed the course of the game for sure.
"It's hard to argue with it but I thought we coped pretty well. We defended magnificently, even when we were down to 12 men. We stifled the way they wanted to play."
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.