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 on making Q3 and qualifying 11th, Bearman produced a blemish-free race to progress to seventh 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.
Source: Autosport