Cudd dived over three minutes from time to not only seal a surprise win but clinch a bonus-point for the Welsh region.
Richie Rees and Rhys Thomas crossed in the first half when the Dragons were reduced to 14 men before Halam Amos and Cudd put Stade to the sword.
Stade have twice reached the Challenge Cup final yet in front of their own fans could still only manage one try, through Herman Bosman.
Vincent Mallet kicked five penalties to keep the game in the balance but the Dragons were ruthlessly clinical with every half-chance in a remarkable display.
That started when O’Brien kicked the opening points in the fourth minute though they lost captain Andrew Coombs to an early knee injury.
Mallet responded with four successive penalties, the last after Boris Stankovich was binned for a reckless high tackle on Nicolas Garrault. Yet the 14 Dragons not only steadied the ship but stunned Stade with two unanswered tries.
Rynard Landman broke clear from a stolen lineout. James Thomas followed, Tom Prydie straightening the attack before Rees stepped through to score.
Two minutes later and Rhys Thomas for their second. Byrne recovered O’Brien’s kick. Matthew Pewtner broke through and Ashley Smith found hooker Thomas unmarked out wide.
Mallet’s fourth penalty levelled the scores before half-time but against the run of play, Dragons caught Stade off-guard once again.
Amos burst off his wing and onto O’Brien’s pass, beating Morne Steyn and Krisnan to score from 40 metres out.
Stade’s response came through their pack that rolled up field before Bosman stretched out to score and Steyn converted.
But O’Brien responded with two more penalties before Cudd dived over late on as Dragons finished the stronger.
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