Perez is just six points behind defending world champion Verstappen after the Mexican’s second victory of the season in Azerbaijan last week.
Aston Martin’s Fernando Alonso, enjoying a strong season, claimed second place and Ferrari’s Carlos Sainz will start in third in an all-Spanish speaking front row of the grid.
Ferrari’s Charles Leclerc crashed out with one minute 36 seconds left, causing a red flag and the decision was taken not to resume the session.
Leclerc, who had clipped the wall earlier in qualification, lost control through Turn Six and span out of control the entry to the next turn, with his Ferrari spinning into the wall.
The premature end to the session left Verstappen, who had topped the final two practice sessions, without the chance to make a final lap to challenge for pole and Leclerc back in seventh.
Mercedes’ Lewis Hamilton, the seven-times world champion, failed to make it out of Q2 and will start down in 13th place on the grid for today’s race.
Duel provides drama
With Red Bull consistently the fastest this season, the duel between their two drivers is so far the source of the most drama.
“I’m enjoying (the battle),” said Perez, “I’m just thinking race by race. Tomorrow (Sunday) is a new opportunity starting from pole, we are the ones that have something to lose,” he said.
Perez said he had been struggling in practice before finding his pace when it counted.
“It wasn’t coming together. I was struggling for balance, confidence,” he said.
“It has been my worst weekend up to qualifying. I was just resetting everything we did, we made a small change into qualifying that really everything became more alive. We put in the lap when it mattered,” added the Red Bull driver.
Verstappen was left frustrated that he had not been able to complete a peak lap before the red flag.
“I was trying to put it on the limit and then I made a mistake and had to abort the lap,” Verstappen said.
“Then you rely on a bit of luck that there is not going to be a red flag, but that can happen on a street circuit. I’m just a bit upset with myself.
“It’s going to be difficult. I made it difficult for myself, so I have to accept that,” he said.
Hamilton has an even more difficult task as his frustrations continued.
“Last run I was at the last of the pack and trying to get the temperatures into the tyres. Everyone slowed into the last corner and I lost all temperatures and couldn’t do the lap,” he said.
“We knew it was very hard and there was a 50:50 chance we could get into Q3 so we need to be better with our timing. It’s done, I’ll try and get my head down tomorrow and see what I can do - 13th to God knows where,” he said.
Kevin Magnusson of Haas was a surprise fourth, ahead of Alpine’s Pierre Gasly and Hamilton’s Mercedes team-mate George Russell.
After Friday’s practice several drivers had complained about the relaid track being slippery and Sainz said it remained a difficult challenge.
“It was a very tricky quali for all,” he said. “I think it’s very difficult to find the right feeling with the tyres around this tarmac, and it was a fight the whole way through and very easy to make mistakes.
“It was getting windier and windier... but in general it was a decent outcome for tomorrow, but today I think we could have been even better,” he said.
Rain is possible today which could add a further element of difficulty.
The debut Miami race attracted plenty of celebrity attention last year and entrepeneurs Elon Musk and Jeff Bezos were both at the track yesterday.
Tesla, SpaceX and Twitter owner Musk watched the practice session from inside Red Bull’s garage while Amazon founder Bezos was with McLaren.
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