Draper won their only previous meeting when he beat Popyrin in straight sets earlier this year on his way to reaching the Qatar Open final, but faced a much tougher battle this time and needed the backing of the home crowd to pull him through.
"There are days when I need that extra support, today was one of them," Draper said.
"Tough match and credit to Alexei, it was a bit up and down, but you guys helped me through that, so thank you."
Draper found himself trailing when the Australian broke to lead 5-4 and served out to win the opening set, but the 23-year-old Briton was unfazed and won four consecutive games to take the second set.
In the decider, Draper held two match points with Popyrin serving, but the Australian rallied to win four unanswered points to make it 5-5.
In the tiebreak, Popyrin broke first, and Draper responded. The Briton made the decisive break to lead 6-5 and smashed his 10th ace of the match to earn a hard-fought victory.
"My family and friends are here and I love being at home and playing in front of them, so it's an emotional moment for me, honestly," Draper added.
Draper will face American Brandon Nakashima after he defeated British wildcard Dan Evans 7-5, 7-6(4) to reach back-to-back quarter-finals on grass after his recent run at the Stuttgart Open.
"My game suits the grass well, so really glad to play well when it mattered today," Nakashima said.
Mensik meltdown, Rune wins
Mensik crashed out after his 3-6, 6-3, 7-5 loss to Bautista Agut, receiving a code violation warning for unsportsmanlike conduct at one stage.
Mensik won the Miami Open in March, beating Novak Djokovic in the final, having defeated Bautista Agut in the first round, but at Queen's, the teenager cut a frustrated figure as the 37-year-old Spaniard fought back after going a set down.
The Czech 19-year-old was 3-1 down in the second set, and after failing to convert a break point, he smashed his racket on the ground, which then slipped from his hands and flew into the stands, where luckily there were plenty of empty seats.
The experienced Bautista Agut now meets Danish fourth seed Holger Rune, who stormed back after losing the opening set to beat American qualifier Mackenzie McDonald 2-6, 6-1, 6-1.
The pair delighted the crowd who remained at the Andy Murray Arena for the late match with some dazzling tennis, with Rune at one stage rescuing a lob from McDonald with a through-the-legs shot from the baseline.
McDonald broke twice in the first set, with the Dane doing likewise in the second, where he also hit 10 aces, and Rune roared into a 4-0 lead in the deciding set to end the American's challenge.