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And Bradley had lock Andrew Farley to thanks as his try sealed the win in in blustery conditions at the Sportsground in Galway.
Bradley's men took a nine-point advantage into the match after last weekend's 27-18 victory in France, but Narbonne had reduced that to just two by halftime.
Connacht veteran Eric Elwood and Craig McMullen traded penalty kicks before visiting fullback Nicolas Nadau scored in first-half injury time to give his side a 10-3 advantage at the break.
Connacht's Mark McHugh eased the home fans' nerves with a try after 66 minutes when some quick passing from close range gave the big centre the space to score, and Farley stormed over the line after a 20 metre burst to cement the win in the closing moments.
In the other match Bath beat Beziers by 26-7 to go forward into the semi-final draw.
Olly Barkley and Simon Danelli were amongst the Bath try scorers.
