Last season's runners-up Edinburgh started their European Challenge Cup campaign with a routine 28-10 victory against Grenoble, while Brive edged past Newcastle Falcons.
Edinburgh lost out 19-13 to Gloucester in last season's final, but Alan Solomons' side got their new European campaign up and running with a win at Murrayfield on Friday.
It took just five minutes for the Pro12 side to score the opening try when Cornell du Preez jumped over after a rolling maul, and two penalties from Samuel Hidalgo-Clyne made it 11-0 at the break.
Hidalgo-Clyne and Grenoble's James Hart exchanged penalties early in the second half, before sustained Edinburgh pressure led to a penalty try just after the hour mark which was converted.
Daniel Kilioni scored a fine interception try to give Grenoble hope with 10 minutes remaining, but Du Preez touched down for his second with the final play of the game to ensure Edinburgh started Pool Five in fine fashion.
In Pool One, Brive were 13-9 victors over Newcastle at the Stade Amedee-Domenech.
Elia Radikedike's fifth-minute try gave Brive a lead which they never relinquished, as Nicolas Bezy contributed eight points with the boot to cancel out three Tom Catterick penalties for the visitors.