The 20-year-old, playing an opponent 15 years his senior and 429 places below him in the world rankings, recovered from a shaky start to win 2-6 6-4 6-3 5-7 6-2 and join third seed Andy Murray in the next round.
The East Yorkshire-based player had come to come through qualifying to reach this stage but nerves - and a partisan crowd - got the better of Edmund early on as Robert broke twice to race into a 5-1 lead, wrapping up the opening set soon after.
However, the Briton regained his composure for the start of the second set and picked up his first break to move 4-1 ahead.
Frenchman Robert pulled that game back in the seventh, but Edmund made the most of three break points in the 10th to level the match.
Another break followed for the world number 121 early in the third set and although Robert again pulled level, the Briton fashioned a 5-3 lead before serving out the set to move ahead in the match for the first time.
Robert’s response was to move 3-0 and then 4-2 ahead in the fourth before Edmund was able to get things back on serve by winning the next two games.
However, when a tie-break seemed likely, the Frenchman pulled another break out of the bag in the 12th to take the match into a decider - much to the delight of the home crowd.
Their jubilation was short-lived, though, as Edmund opened up a 4-0 advantage with some confident shots.
Robert managed to win the next two games to give himself a fighting chance, but was broken again in the seventh to allow his opponent to serve for the match.
The nerves threatened to take over once more as Edmund found himself facing another break point, but fought off the danger before sealing the win and a second-round meeting with Australian 29th-seed Nick Kyrgios.
Should he come through that test, then Murray is likely to be the player awaiting him in round three.