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Second seed Alcaraz broke twice in each of the first two sets and seemed to be cruising but a dogged Dzumhur refused to roll over and pinched the third set.
The 33-year-old Dzumhur had failed to get past the first round of qualifying at Roland Garros in his last four attempts, but he broke to open the fourth set and give Alcaraz greater cause for concern.
Alcaraz's frustration grew when three break points slipped beyond his grasp the very next game. He eventually got back on serve at 3-3 though before another break put him within sight of victory.
The Spaniard couldn't close it out initially but then earned a pair of match points on Dzumhur's serve, finally getting across the line at the second attempt to end his rival's spirited effort.