Perez was the first of the front-runners to stop at the end of lap 23 but for the second weekend in a row, Red Bull experienced a slow stop, this time with the rear-left tyre sticking when being removed.
It was this five-second hold that the Mexican believes cost him the chance to better Hamilton for the runner-up spot.
"I gave my best at the start," insisted Perez.
"Unfortunately, we had a bit of a bad stop which prevented us from undercutting Lewis and overtaking is difficult so once I was behind him, it was difficult to follow so I had to stay in third."
Russell was left flat-footed by an attack from team-mate Hamilton, with Perez pouncing and making his move around the outside of turn four.
Asked if this was crucial to his race, Perez replied: "Yeah, definitely.
"Because of how difficult overtaking was today, that was key.
"I thought we would be a bit closer but it didn't work out for us today but still, it is a good podium in front of this crowd.