The French side led for almost all of the game after scoring in the third minute, but Connacht's kicker Miah Nikora expertly kept them within touch with the hosts.
Saimoni Vaka gave Bayonne the lead when he broke free of several tackles to score a try in the third minute, and Matthieu Ugalde followed him across just 15 minutes later.
Brade got the Irish side back into the game with a simple try, before a couple of Nikora penalties sent them in at half time just one point adrift.
Winger Pierre Sayorse scored Bayonne's third try straight after the break, while they continued to add to their score through penalties.
Nikora remained accurate by scoring two further penalties, before scrum-half Brade crossed for his second try in the 76th minute to snatch his side the win.