Sergio Perez seizes chance as car 'comes alive' to claim pole
The 33-year-old Mexican rose to the challenge after defending double world champion Verstappen was eliminated by a driveshaft failure in Q2 after dominating all three practice sessions.
Perez clocked a best lap in one minute 28.265 seconds to beat Ferrari's Charles Leclerc, who faces a 10-place grid penalty, by 0.155sec in the final runs of a closely fought qualifying session.
It was his second successive pole at the Jeddah Street Circuit.
"It was tricky in that Q3," said Perez. "Especially getting that second lap and nailing that lap you really feel the Formula One cars coming alive in this place so maximising that lap was really important.
"It's a shame for Max though because he has been really strong the whole weekend. Hopefully tomorrow we can have both cars up there as you never know with these cars and reliability issues can hit you at any time."
Verstappen's car had eased to a halt in Q2, leaving him to limp home to the Red Bull pits where he retired. He will start from 15th on the grid.
"It happened without any warning and we have had no trouble before," said Verstappen. "It's the first time it's happened. And it's very annoying because we were having such a good weekend.
"It will be tricky now to get to the front so we have to concentrate on points, but anything is possible at this track."
Team chief Christian Horner said Red Bull were concerned by the failure and the team would "get to the bottom of it".
The team had changed gearbox parts in both cars.
He said: "It wasn't a concern going into the race, it was just part of our strategy of introducing parts into the pool, but something has obviously happened there and we need to get to the bottom of it and understand it and try and ensure that it doesn't happen tomorrow.
"It's a tough race here and it's about staying out of trouble. There's going to be plenty of action, plenty of carnage and it should provide an exciting race hopefully.
"It's a great opportunity for Checo and it's important to convert that. So, we are going to have two different races going on in the Grand Prix tomorrow."