The 28-year-old, who converted each of those poles into victories, was not expected to be in the mix for pole with Mercedes dominant under Formula 1's sweeping new regulations and Ferrari and McLaren also ahead of Red Bull.
However, he failed to even make the top-10 shootout, with Racing Bulls rookie Arvid Lindblad grabbing the final spot.
Verstappen's teammate Isack Hadjar made it through to the final phase of qualifying and will start eighth.
“Well, the car never turns mid-corner, but at the same time this weekend also again, we just oversteer a lot on entries,” Verstappen told Sky Sports.
“We thought we fixed it a little bit ... but now in qualifying it was again, for me, undriveable.
"Also I'm driving with a different aero package this weekend, but it seems like that's not working, so that's also not very good,” he added.
Verstappen, the winner of 71 races, is eighth in the championship on eight points from two races. Mercedes driver George Russell leads the standings on 51 points.
Verstappen raced from 20th to sixth in the season-opening race in Australia but failed to score points in the Saturday sprint in China and retired from the main race.