Alcaraz, who became the youngest man to complete a career Grand Slam with his victory in Melbourne, saw off the 30th-ranked Rinderknech in two closely contested sets.
"It was really difficult. Arthur is a really dangerous player. Nobody wants to play against him in the first round," said Alcaraz.
"I'm happy with the level. I'm happy that I got through difficult moments in the match. I'm happy that I stayed calm and positive, and played great tennis."
The world number one will face another Frenchman in round two, 60th-ranked Valentin Royer.
Alcaraz took the first set after breaking in the fifth game but faced dogged resistance from Rinderknech, who brought up his first break points of the match with the top seed serving to force a tie-break in the second set.
The Spaniard saved both and then closed out victory in the tie-break, ripping a forehand down the line on match point.
Alcaraz's chief rival, Jannik Sinner, is on the other side of the draw. No player other than Alcaraz or Sinner has won a tour-level title at an event featuring both of them since Andrey Rublev lifted the trophy in Madrid in May 2024.
Rublev booked his place in the second round with a straightforward 6-4, 6-3 victory over Dutchman Jesper De Jong.
Alcaraz, who lost in the quarter-finals on his Qatar Open debut a year ago, could face one of three former champions in the draw, Karen Khachanov, in the last eight if he gets past Royer on Wednesday.
Khachanov and Tsitsipas progress
Seventh seed Khachanov needed three sets to overcome Japanese lucky loser Shintaro Mochizuki, while Stefanos Tsitsipas downed Tunisian wild card Moez Echargui to line up a meeting with 2023 winner Daniil Medvedev.
Jiri Lehecka, who knocked Alcaraz out of last year's tournament, swept past American Jenson Brooksby in straight sets in his opening match.