Alcaraz survives Queen's scare as Murray's winning streak ends
Alcaraz was playing only his seventh match on grass and his first outside of Wimbledon as he prepares for the third Grand Slam tournament of the season next month.
Rinderknech, ranked 83 in the world, only found out he was playing the Spaniard hours before the match after Frenchman Arthur Fils, Alcaraz's scheduled opponent, withdrew from the tournament.
But the 27-year-old almost stunned world number two before Alcaraz edged through 4-6, 7-5, 7-6 (7/3) in just over two-and-a-half hours.
"It has been a really tough match, he has played four or five matches on the grass and this was my first one," said Alcaraz.
"It was really difficult to adapt my game to the grass. For me it's tough to play here, but Queen's is a tournament I really wanted to play."
Alcaraz was playing his first match since losing to Novak Djokovic in the semi-finals at the French Open and is hoping to improve once he gets accustomed to the conditions.
"I couldn't play tennis on grass at home but I practised the movement on grass. I practised here a little bit before the tournament and I felt really good, even though it was the first match.
"I would say I'm going to be better in the next round."
Alcaraz will face Czech Jiri Lehecka in the second round.
Murray's winning streak ends
Murray had been on a 10-game winning streak after winning Challenger Tour events at Surbiton and Nottingham.
But the former world number one found the step up in quality to hard to handle as De Minaur eased to a 6-3, 6-1 victory.
Murray had been hoping to rise into the seeds for the Wimbledon draw, but that is now unlikely as he started the week ranked 38 in the world.
"I was so excited to play Andy on Centre Court at Queen's. I'm super proud of how I played and it couldn't have gone any better for me," said De Minaur.