Mahindra Racing's Bruno Senna came home fifth while Karun Chandhok was forced to retire with a broken rear suspension in the street race held yesterday.
Da Costa, driving for Amlin Aguri, kept clear of trouble and slowly made his way into the lead as his rivals fell by the wayside. Both polesitter Sebastien Buemi and championship leader Lucas di Grassi retired with suspension failures at the fast Turn 8 and 9 chicane.
Venturi's Nick Heidfeld looked favourite to win in the closing stages but a drive-through penalty for speeding in the pitlane scuppered his chances.
Prost managed to steer clear of the battle between Daniel Abt and Jaime Alguersuari in the closing stages, an over-optimistic move by the Audi Sport ABT driver sent the Spaniard into a spin. Abt was out on the spot but Alguersuari managed to limp his car home in fourth.
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Piquet picked up his second consecutive podium despite being held behind the red light and losing a lap in the pitlane during the safety car period.
The safety car was deployed after Chandhok also fell victim to broken rear suspension at the Turn 8 and 9 chicane. Piquet wasn't the only driver to be effected by the confusion as a number of drivers had to pass the safety car to unlap themselves.
Senna, who was one of the FanBoost winners, finished fifth ahead of a feisty Jean-Eric Vergne who damaged his nosecone in the fight for the final podium spot.
Da Costa's team-mate Salvador Duran clinched four points in eighth ahead of Heidfeld in ninth. Dragon Racing's Oriol Servia rounded out the top-10 positions while Formula E debutant Marco Andretti came home in 13th place.
Before the race all the drivers and teams together with the City of Buenos Aires, Argentinean authorities and the FIA President Jean Todt gathered on the grid for a minute silence to pay respect to the victims of the dramatic terrorist attack which happened in Paris earlier this week.