A late Alex Lewington try secured the win for Irish to move them five points clear of the French side at the top of pool ahead of their return match next Saturday.
Tom Fowlie had scored for Irish in the first half and former England fly-half Shane Geraghty kicked 14 points to put the Aviva Premiership side in charge of their group.
Olivier Missoup scored a second half try for the visitors and Jerome Porical kicked eight points in a game that saw five yellow cards.
Fergus Mulchrone, Nic Rouse and Mike Mayhew all saw yellow for the home side while Stade lost Djibril Camara and Laurent Sempere to the sin bin.
Newcastle Falcons kept themselves hot on the heels of former Heineken Cup champions Brive in the race to be crowned Pool Three winners with a 37-15 try bonus-point win over Rugby Calvisano at Kingston Park.
In a match in which yellow was also very much the colour of the day, French referee Cedric Marchat sent eight players to the sin bin - six of them from the Italian club.
Flankers Maxime Mbanda and Nicola Belardo, centre pairing Tommy Castello and Andy Vilk, tighthead prop Lorenzo Romano and scrum half Simon Piconealso all saw yellow for Calvisano while Newcastle lost Will Welch and Joel Hodgson in the second half.
Newcastle wing Zach Kibirige got their first try after 24 minutes and full-back Noah Cato their second two minutes later.
That suggested a deluge of tries for the home side but instead they had to wait until the 50th minute to bank the bonus point, number eight Chris York getting try number four after the Falcons had earlier been awarded a penalty try.
Left wing Michele Visentin and number eight Braam Steyn did cross for the visitors but Cato collected his second and the Falcons' fifth in the 66th minute.