Trailing 20-2 early in the second period, Brett Hodgson’s inspired side got back to 21-18 in the 72nd minute after tries from Mahe Fonua, Danny Houghton and Cameron Scott.
And, in the set after Scott’s effort, Jake Connor’s perfect 40/20 looked to have put them in sight of the chance of a famous win.
However, there was heartbreak for Hull full-back Connor when, under pressure from the Saints defence, he tried another defence-splitting pass for Fonua only to see Regan Grace intercept, race 80m and score at the other end.
To add salt to the wound, it was Saints who instead then scored back-to-back tries, Connor failing to get off the ground as Kevin Naiqama rose to palm down a high kick for Lachlan Coote to add their fifth try of the afternoon and confirm their place in next month’s Wembley final.
However, arguably the most crucial point of a fascinating game, played in sweltering conditions at Leigh Sports Village, came much earlier in the 25th minute.
In-form centre Griffin had beaten the first Saints defender coming away from near his own line when he grimaced in agony with no one near him, went lame and dropped the ball before falling to the ground, clutching his right ankle.
Saints scrum-half Theo Fages immediately picked up the ball and scrambled over to score from 10m out.
Hull’s fans were incensed. Griffin, punching the ground in obvious pain as he awaited medical attention, was helpless.
Yet, as harsh as it may sound, it could be argued Frenchman Fages simply played to the whistle; referee Liam Moore certainly did not stop the game.
Maybe he should have done as it was clearly a serious injury but Hodgson had no qualms with it afterwards.
The upshot of it all was, once Coote tagged on the conversion, the Super League champions were ahead 14-2 and always likely to progress as they seek to win the Cup for the first time since 2008.
It was hard on Griffin, too, the 30-year-old who must have been in contention for an England debut against the Combined Nations All Stars later this month and had his heart set on playing in the World Cup.
It was always going to be tough for his side to recover from there.
Yes, they had a centre on the bench in the shape of Scott, so there was no need for a major reshuffle.
But Hodgson had lost one of his main strike players and gone 12 points down in the process.
It was especially hard for Hull to take given they had been the better side for the opening 15 minutes, when their opponents looked disjointed to say the least.
Morgan Knowles spilled in contact, Louie McCarthy-Scarsbrook fumbled playing the ball and when there was yet another error handling, Saints were caught offside to allow Marc Sneyd’s boot to give Hull the first points.
Nevertheless, that was all FC could muster from that initial positive spell.
In contrast, when they erred for the first time, Chris Satae losing the ball in the 16th minute, Saints reacted with alacrity, scoring in the next set.
Fages delivered the killer blow with his bullet cut-out pass seeing Grace get over one-handed in the tightest of spaces.
Coote slotted the first of his five goals, the ball ricocheting in off a post.
Jack Welsby got their third try in the 49th minute, getting on the outside of Carlos Tuimavave and, though tackled, keeping his ball-carrying arm off the ground to enable a second bid.
Coote converted but, from the short restart, Saints’ Lewis Dodd clearly blocked the onrushing Connor, to give Hull some much-needed position and possession.
This time, they made the most of it, Sneyd, Josh Reynolds and Connor linking smartly to the right for Tuimavave to smuggle Fonua over.
The big winger was unfortunate not to get a second soon after, just failing to hold on to Connor’s chip as his side attacked that right corner once more on the last tackle.
Fonua looked to have been nudged before attempting to collect but Saints - who also coughed up possession in the next set - were clearly rattled.
It was no surprise, then, that Fages slotted a drop goal the next time they had a chance in the 65th minute.
Still, the tactic backfired somewhat; Hull, once more, gained possession from a short restart and, when McCarthy-Scarsbrook was then penalised for a ball steal, Hodgson’s side scored a fine try from a set move.
Houghton attacked the short-side from the tap and claimed Tuimavave’s offload to dive over, Sneyd’s conversion seeing them trail just 21-14.
Although Houghton’s misplaced pass in the next set was picked off by Fages - who else? - to relieve the pressure, Hull defended their own line and then went on the attack once more.
This time it was Scott who came to the fore, doing wonderfully to gather Connor’s flicked-on pass and then force his way over.
But there was no final push. Only the one from Saints.
Hull FC: Connor; Fonua, Tuimavave, Griffin, Swift; Reynolds, Sneyd; Sao, Houghton, Satae, Savelio, Ma’u, Lane. Substitutes: Bowden, Brown, Scott, Fash.
St Helens: Coote; Makinson, Naiqama, Welsby, Grace; Lomax, Fages; Walmsley, Roby, McCarthy-Scarsbrook, Thompson, Mata’utia, Knowles. Substitutes: Amor, Batchelor, Dodd, Norman.
Referee: Liam Moore (Wigan)