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The Spanish star came through 6-1, 3-6, 6-1, 6-2 against his 112th-ranked opponent and will face Canada's Denis Shapovalov for a place in the fourth round.
Alcaraz, 20, is bidding to add the Roland Garros title to the US Open he won in 2022.
"I'm very happy with my level. The wind meant conditions were really difficult so I had to adjust as much as possible, I was very focused on each shot," said Alcaraz.
Victory on Wednesday gave Alcaraz a 22nd win from 24 on clay in 2023, a run which has yielded titles on the surface in Buenos Aires, Madrid and Barcelona.
"Taro was playing great. He is a really tough player and this year he has won matches against great players," said Alcaraz who is top seed at a Grand Slam for the first time.
On Wednesday, he fired 46 winners past New York-born Daniel.
Shapovalov, the 26th seed, saw off Italy's Matteo Arnaldi in four sets to reach the third round in Paris for the first time.