The British 18-year-old got called up by Ferrari on Friday morning after Sainz was sidelined with appendicitis, meaning he had just final practice to prepare ahead of qualifying in Jeddah.
After just missing out in Q3 and qualifying 11th, Bearman produced a blemish-free race to progress to seventh place at the chequered flag, ahead of fellow Britons Lando Norris and Lewis Hamilton.
While Bearman expects Sainz to return to action next time out in Australia, and he will resume his Formula 2 campaign with Prema, he acknowledged this was his major chance to prove he is capable of competing in F1 in the future.
"I don't know what else I can do, because I don't think I'll be in F1 for the rest of the year," Bearman said.
"That was my goal, to do a great showing this weekend, I think I did a decent job, so that is alright.
"That is all I can do, keep pushing in F2 and cross my fingers."
Bearman is scheduled to undertake six FP1 sessions for Haas during the 2024 season, starting at the Emilia Romagna GP at Imola in May, in an agreement that was expected to be part of his F1 evaluation and audition to join the US squad from 2025.
Both Kevin Magnussen and Nico Hulkenberg's Haas F1 contracts expire at the end of the season.
"I have a lot of FPs with Haas this year so I am looking forward to building up a relationship with them and gaining more miles in the car," Bearman added.
"Hopefully a door can open, that would be fantastic."
Assessing his Saudi Arabian GP performance, Bearman was content with his showing given his limited experience and preparation as he focused on completing a mistake-free race.
"The car was flying today so that is obviously a big bonus," he said. "But I think we executed a clean race, no mistakes, and that is exactly what we were looking for so I am happy with my performance."
Ferrari has not made a decision on if Sainz will be back in action in Australia, as the Spaniard has two weeks to recover from his appendicitis operation.
The two-time grand prix winner visited the Jeddah paddock after his surgery and watched the race from the Ferrari garage.
