Aston Martin's 41-year-old double world champion Fernando Alonso, who had hoped for his first pole since Germany 2012 with Ferrari, will join the Mexican on the front row for Sunday's race in Jeddah.
Ferrari's Charles Leclerc qualified second fastest but has a 10-place grid penalty that will drop the Monegasque to 12th place.
Perez's best time of one minute 28.265 seconds around the floodlit Corniche street circuit was more than a tenth quicker than Leclerc's effort.
"You really feel the Formula One cars coming alive in this place and maximising that lap was very important," said Perez of what was also his second career pole.
"It's a shame," he added of Verstappen's early departure. "Max has been really strong the whole weekend so hopefully tomorrow we can have both cars up there."
Alonso will be the first Aston Martin driver to start a Grand Prix on the front row since Britain's Roy Salvadori at Silverstone in 1959.
"Qualifying was our weak spot in Bahrain but we performed very well so let’s see what we can do. It is amazing," said the Spaniard.
"We are confident, we feel very strong. The strongest part is the long-run pace so it should be better on Sunday."
Mercedes' George Russell and Ferrari's Carlos Sainz will share the second row.
Verstappen, last year's winner in Jeddah, had dominated all three practice sessions and was fastest in the first phase of qualifying before it all went wrong.
"I have a problem," he told the team over the radio.
"It's almost not accelerating," added the Dutch driver as he nursed the car back to the pits and was pushed back into the garage before stepping out with just over six minutes of the second session remaining.
"It will be a bit more tricky to get to the front but it will be all about scoring points," said Verstappen.
"Of course, I would have liked to have started further up the road but there is nothing I can do now. Let's stay realistic -- it is going to be tough but we have good pace so let's move forward."
McLaren's woes continued after a tough season-opener in Bahrain, with Lando Norris set to start 19th following contact with the wall that damaged his car in the first phase of qualifying.
Australian rookie teammate Oscar Piastri made it through to the second phase and will start eighth.
Both the AlphaTauris also went out at the first hurdle, with Yuki Tsunoda qualifying 16th and Dutch rookie teammate Nyck de Vries spinning on his way to 18th.
The two Williams also failed to make it through to Q2, Alex Albon qualifying 17th and U.S. rookie Logan Sargeant last of all after he exceeded the track limits.
