The British No. 1, who beat Novak Djokovic and David Goffin en route to the quarter-finals in Madrid last week, was forced to battle for his 6-3 3-6 6-3 success.
Having cracked the world's top 20 for the first time this week as a result of his Madrid heroics, Edmund was always going to be full of confidence on a surface he regards as one of his favourites.
But Tunisian Jaziri was never going to prove a pushover, with the talented but erratic world No. 64 having reached the final of the Istanbul Open last month.
Edmund was forced to save break points in his opening service game but hit back to break Jaziri for a 4-2 lead and maintained his advantage to seize the first set.
Jaziri was not going to roll over, and broke Edmund in the fourth game of the second set, proceeding to fight off two break points on his own serve before levelling the match.
Edmund kept his nerve to grab the solitary break of the decisive set and move through to round two, where he will come up against either Andreas Seppi or 16th seed Lucas Pouille.