Wimbledon champion Alcaraz toiled for three hours in stifling heat on Thursday to edge past fellow Spaniard Jaume Munar in a titanic scrap and afterwards admitted he did not know how he was still standing.
It was far more straightforward against unseeded Rinderknech, the player he also beat in his opening match two years ago when he went on to win the title at the prestigious Wimbledon warm-up event in west London.
The 22-year-old converted each of his two break points as he extended his sequence of match wins to a career-long 16.
"I could be better (physically). Honestly I was thinking I was going to feel much worse than I did today," five-time Grand Slam champion Alcaraz said on court.
"I'm glad it was only one hour 20 minutes today."
Alcaraz will face Spanish veteran Roberto Bautista Agut in the semi-final, after he stunned fourth seed Holger Rune, 7-6(5), 6-7(4), 6-2.
Meanwhile, Draper not only kept himself on track for the prestigious ATP 500 title, but also secured the win that guarantees a top-four seeding at Wimbledon, where he will be the big home hope.
The 23-year-old was fully tested by world number 32 Nakashima and his powerful game began to misfire as he lost the second set on a baking hot day on Andy Murray Arena.
But Draper settled back into the groove in the deciding set and broke serve in the seventh game with one of his trademark forehand pile drivers.
The left-hander had to save a break point when serving for the match at 5-4 but recovered to seal victory and set up a clash with Czech Jiri Lehecka who had earlier beaten Britain's Jacob Fearnley 7-5, 6-2.
"It means the world to me to reach the semi-finals here," Draper, bidding to become the first British player to win the title at the prestigious west London club since Murray won it for the fifth time in 2016.
While being ranked fourth means Draper will avoid either defending champion Alcaraz or world number one Jannik Sinner until the semi-finals at Wimbledon, Draper shrugged off the significance of that when asked about it.
"I kept getting asked about that by the journalists but I've got to reach the semi-finals first," he said.
"But it is an incredible position, I was at Wimbledon last year ranked 40 so to go there as fourth seed is a testament to all the hard work my team has done so very proud of that."