Ferrari's pole-sitter Charles Leclerc held off the Aston Martin of Fernando Alonso to complete the podium, taking a much-needed third.
"Vamos!" exclaimed Perez on the team radio after making it a hugely profitable weekend in Baku where he also claimed Saturday's sprint race.
Red Bull's roll through 2023 after four races now reads four wins plus Saturday's sprint, three poles, and three 1-2s.
Verstappen leads the drivers' championship by just six points ahead of the fifth round of the record 23-race season in Miami.
Perez's engineer told him on the team radio he was the "king of the streets" and with justification as five of his six F1 wins - not counting Saturday's sprint - have come on street circuits, two in Baku alongside victories in Saudi Arabia, Monaco and Singapore.
The Mexican believes he can challenge Verstappen for the title, and was glad to keep the pressure on his team-mate
"It really worked out today for us (me), we managed to keep the pressure on Max," he said.
I think it was very close between us, we pushed to the maximum, we both hit the wall a few times but we managed to keep him under control."
Verstappen led his team-mate early on, but ended up behind when a Safety Car came out just after he made his first pit-stop, allowing Perez to pit without losing much time to the rest of the field and rejoin ahead of everyone.
The Dutchman wasn't angry with his team for bringing him in when they did though, saying it was just a case of bad luck.
"Checo (Perez) got a little lucky with the safety car there but it’s a long season and very good points today. Great team result," Red Bull team boss Christian Horner told Verstappen at the finish.
"It is what it is. We wouldn’t know the safety car could come out there. Good team result," replied the Dutch driver.
Leclerc conceded that for the time being at any rate Red Bull were in "another league once it comes to the race".
"Over 51 laps it was not possible, they have so much more pace than we do. Everyone is working flat out to understand what we can do in the races to close the gap."
Carlos Sainz finished fifth behind Alonso with seven-time world champion Lewis Hamilton sixth for Mercedes and Aston Martin's Lance Stroll seventh.
George Russell was eighth ahead of McLaren's Lando Norris and AlphaTauri's Yuki Tsunoda collected the final point.
There were worrying scenes on the final lap with Alpine's Esteban Ocon coming in for his sole stop and sending mechanics and photographers who were walking out into the pit-lane scattering out of his way.
