This was a match Braga dominated for long spells in the first half, prompting Rodgers to change his formation after the break in the hope that a back three and extra men in the middle might solidify his side.
The Scottish champions showed improvement thereafter and thought they had equalised when Kelechi Iheanacho calmly slotted home after a mistake from the Braga backline.
His celebrations were ended by referee Tobias Stieler blowing for handball, a decision backed up by a lengthy video assistant referee (VAR) check, although footage appeared to show the ball did not hit a hand.
Braga then had goalkeeper Lukas Hornicek to thank for two super stops as Celtic ramped up the pressure in search of an equaliser.
First, he leapt to his left to keep out a Kieran Tierney header, before hurling himself to the other corner to tip away Sebastian Tounekti's curling effort which looked destined for the net.
After the change of formation, Rodgers then threw on several substitutes - including Reo Hatate, Arne Engels and James Forrest - but no-one could conjure the quality to find the goal Celtic needed.