The pair now have two wins apiece after four rounds of a record 23-race season, with Red Bull in a league of their own.
Perez is six points behind double world champion Verstappen due to finishing only fifth in Australia -- a race the Dutch driver won -- after problems in qualifying.
"Having three kids at home, I wouldn't be travelling around the world if I didn't believe that I can be a world champion and I'm working towards that," Perez told Sky Sports TV after Red Bull's third 1-2 of the season.
"There is so much you can talk outside the car, it's important to deliver on the track and I think without the issues we had in qualifying in Melbourne we should be leading the championship.
"So definitely there is everything to believe that we can do well this year."
Perez won in Saudi Arabia and finished second in the Bahrain season opener.
Sunday's win in Baku was helped by the safety car but he still made the most of the opportunity to become the first driver to win there twice.
"We are just pushing each other massively and I think this year is about not making those mistakes," said Perez, who also won the first sprint race of the season on Saturday in Baku.
"I am fighting for it," the Mexican driver said of the championship. "But I also know it's a long road ahead."
While the number one on Verstappen's car reflects a de facto status within the team, Perez is keeping him in his sights with his best ever start to a season.
Perez has yet to win more than twice in a season, while Verstappen is a 37-time winner but has raised his game.
"I think we delivered under pressure massively because every single session was so critical in this [sprint weekend] format," Perez said. "It was really important to deliver when it mattered.
"Yesterday we executed a perfect short race and today once again we executed a tremendous race."
Team boss Christian Horner said Perez, who has had all but one of his six career wins on street circuits, had been "sensational."
"He's always been outstanding at this track," he added. "He absolutely excels here. He got a little lucky with the safety car today but then he converted that and pulled a gap and he had the pace and really controlled the race.
"A phenomenal drive by him all weekend."