George Russell qualified second for Mercedes with McLaren's Lando Norris and Oscar Piastri third and fourth respectively. Ferrari's Charles Leclerc will start fifth with teammate Carlos Sainz seventh.
McLaren lead Ferrari by 30 points, after a one-two in the earlier sprint race, and can clinch their first constructors' title in 26 years on Sunday if results go their way.
Verstappen secured his fourth successive title in Las Vegas last Saturday and the Qatar pole was his first since Austria at the end of June.
"That was amazing, the turnaround. Great job guys," Verstappen said over the radio.
"I didn't expect this, he added afterwards.
"We changed a bit on the car but I never thought it would make such a swing in performance, so that's promising. I hope it lasts tomorrow in the race.
"It just felt a lot more stable over one lap and that's exactly what we need."
Seven-time world champion Lewis Hamilton qualified sixth for Mercedes with Fernando Alonso eighth for Aston Martin and Verstappen's struggling teammate Sergio Perez ninth.
Kevin Magnussen completed the top 10 on the grid for Haas, who are fighting a midfield battle for sixth place with Renault-owned Alpine and Red Bull's RB.