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The British No 1 – who rose to a career high No 19 in the ATP rankings on Monday – needed more than two hours to dispatch the 34-year-old, with just one break deciding each set.
It was a decidedly tighter affair than their match on the red dirt in Morocco last month, which Edmund won with consummate ease 6-2 6-1.
Up next for Edmund is Lucas Pouille or Andreas Seppi – while Alexander Zverev looms as a potential third-round opponent.
Edmund saved two break points in the opening game of the match before moving 4-2 in front and seeing out his two remaining services to take the set.
However, momentum swung Jaziri’s way when the world No 64 broke Edmund in the fourth game of the second set before saving two break points a game later.
The Tunisian then rallied from 0-30 behind when serving for the set to level up the match and take it to a decider.
Edmund dug deep to hold serve before breaking Jaziri in the fourth game, with the 23-year-old going on to close out the match after two hours and seven minutes on court.
Dame Katherine Grainger, Helen Glover & Greg Searle