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Despite the Super League leaders looking set to be overturned by second-placed Warrington, Radford's side came from behind to grab a late 12-10 win.
"That's the togetherness we have in this team," he told BBC Sport.
"We have the ingredients there to go on from this. We have the foundations in place to go and achieve more."
Hull, who have already qualified for the Super League play-off semi-finals, still have to host both Warrington and third-placed Wigan, as well as visiting fourth-placed St Helens before determining who might line up at Old Trafford for the Grand Final on 8 October.
Radford added: "There's the League Leaders' Shield to still play for and then we could be only 80 minutes away from a Grand Final."
The same two goals are also still very much in the sights of Tony Smith's Warrington.
"It will go down as one of the better finals for many a year," said the Wire boss, after his club suffered their first defeat at the rebuilt Wembley and missed out on a fourth Challenge Cup victory in eight seasons.
"We'll go away and lick our wounds, but we won't sulk for too long. We're a proud club and we've still got a great opportunity to win two other trophies."
Even in a sport that is full of ups and downs in so many contests, it is an impressive statistic that Saturday's final was the 14th time that Hull have come from behind to win this season.
"It's history-making for us and we should feel rightly proud," said Radford. "There's been a lot of past Hull teams with some tremendous players who have not achieved what we have.
"And it was great for Jamie Shaul to get the winning try. He's a local lad like me, so he gets a lot of stick when we lose. But he can now go and walk his dog with pride."
Modest Hull scrum-half Marc Sneyd played down his own contribution to Hull FC's late rally after being named man of the match.
"I was surprised I'd won," he said, suggesting that he was expecting hooker Danny Houghton to pick up the Lance Todd Trophy.
His coach Radford also had praise for Houghton, who made a try-saving tackle on Ben Currie with two minute remaining, adding: "He would not look out of place in the England team."
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But Hull skipper Gareth Ellis, finally successful in his third final, insisted that Sneyd fully deserved his individual accolade.
"Don't let him kid you he was not a worthy winner of the Lance Todd," Ellis said. "It was a hot, sapping day and when he put that 40-20 in (before Mahe Fonua's try) you could see some Warrington heads drop.
"Warrington were the better team for 60 minutes, but Marc has been quality with his kicking all season and he produced it when it mattered. He was superb."
Proud Hull FC boss Lee Radford predicts there is more to come from his side after the Black and Whites' first Challenge Cup final win at Wembley.