It is first time that Force India has celebrated two podium places in a single Formula One season.
Perez, who had also finished third in Monaco in May, had to start from the seventh position on the grid despite earning P2 in Saturday's Qualifying due to a five-place penalty imposed on him owing to a gear box change during FP3.
However, the Mexican scorched the street circuit in the inaugural F1 race in Baku to take third spot behind winner Mercedes' Nico Rosberg and Ferrari's Sebastian Vettel.
Perez now has four podiums to his credit for Force India. Last year, Perez had claimed third place in Russian Grand Prix while in 2014 he was third in Bahrain Grand Prix.
The best performance for the team remains the one achieved by Giancarlo Fisichella in 2009 when he finished second in the European Grand Prix in Valencia.
Force India collected 17 points from the race to take its tally to impressive 59, further consolidating its fifth position in the constructors standing.
Perez said: "Yesterday, when I hit the wall in qualifying I was very frustrated with myself, but the team did a great job to repair the car.
"And today, after starting from P7, it was a fight all the way.
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