Perez led Verstappen home after a tepid Azerbaijan Grand Prix, but his afternoon nearly came undone when on lap 34 he clipped the outside wall on the entry to Turn 15.
The Mexican came away unscathed but afterwards admitted his error led to a hard thump on the right-front corner of his Red Bull RB19.
"Really hard," he said about the hit. "I had a bit of luck, especially with the front right that it didn't blow up."
Starting from third, Perez quickly dispatched Ferrari's Charles Leclerc and then took the lead when Verstappen pitted on lap 10, just before the safety car came out for Nyck de Vries' stricken AlphaTauri.
Perez benefitted from the neutralisation to complete his only pitstop and remain in the lead.
And after controlling the restart the Mexican put in a composed drive on the Baku City Circuit, never coming under sustained pressure from Verstappen.
"It really worked out today for us. We managed to stay in the DRS train and we managed to keep the pressure on Max," Perez said.
"I think we had a better deg on that first stint, it was looking good already from that side.
"Then the safety car came and bunched everyone up. So, it was again another race on the hard tyres.
"I think it was very close between us. We pushed to the maximum today, we both clipped the wall a few times we were pushing out there.
"The way Max pushed me throughout the race was really hard but we managed to keep him under control."
Perez became the first repeat winner in Azerbaijan and after also winning Saturday's sprint he has closed the gap to Verstappen to six points, putting him more firmly in the hunt after four of 23 races.
After falling to third Verstappen quickly passed Leclerc on the restart but then started struggling with the balance of his car and never got close enough to threaten his team-mate.
"Of course, the safety car was a bit unlucky and then I had to push again, I tried to stay very close to try and get into the DRS," he explained.
"The tyres were overheating a bit because of that, trying to follow.
"But also the balance, I was struggling to be really consistent, so I was playing around with the tools a little bit.
"And I think once I got that sorted, I would say the last 10 laps were actually quite good again, but just a little bit too late."
