The unseeded Canadian pulled off another US Open upset when he knocked out the eighth-seeded Bautista Agut 7-5, 2-6, 4-6, 6-3, 6-2 on Saturday.
Pospisil, who eliminated fellow Canadian and 25th seed Milos Raonic in the second round, has already spent nearly 10 hours on court at Flushing Meadows and advanced in emphatic fashion after smashing 71 winners and 19 aces.
Pospisil lost his focus after taking the opening set with a late break, as he allowed Bautista Agut back into the contest, but got a new lease of life after a medical time-out in the fourth set and roared to victory.
Serena Williams survived a third-round scare to topple 2017 champion Sloane Stephens 2-6, 6-2, 6-2 and keep alive her bid for a record-equalling 24th grand slam title.
A sure-footed Stephens pounced early as third seed Williams struggled with her first serve and committed 13 unforced errors in the first set, claiming an early break for a 2-1 lead.
The 26th seed kept Williams on the run as she saved two break points, with the 38-year-old struggling to find her rhythm.
But the momentum dramatically shifted in Williams’ favour in the second set. She opened up a 4-2 lead as she cleaned up her game and kept psyching herself up in the absence of the legions of fans who regularly flock to Flushing Meadows. The stands are empty this year due to the coronavirus pandemic.
“It was intense, I have to say,” Williams, chasing a seventh title at the New York major, said in an on-court interview. “It always brings out the best in my fitness when I play Sloane.”
She sent down a dozen aces, including one on her second serve.
Williams, who next faces Greek Maria Sakkari, played the match in front of her daughter, three-year-old Olympia, and said she hoped “that she saw her momma fighting”.