The Virgin Racing driver now has 43 points, to Belgian Jerome d'Ambrosio's 41.
D'Ambrosio crossed the line ninth for Mahindra and then moved up to eighth, after Argentina's Jose-Maria Lopez received a penalty, which would have kept him a point clear of Bird.
However, he was subsequently dropped to 10th after being handed a five-second penalty for speeding under yellow warning flags.
Bird took over at the front with 11 laps to go after Swiss rival Sebastien Buemi, the 2015/16 champion, crashed out while leading from pole position on the fenced-off Santiago street circuit.
D'Ambrosio's rookie team mate and fellow former F1 driver Pascal Wehrlein finished second, for his first Formula E podium, with Audi Sport's Daniel Abt third on a sweltering day with track temperatures of 38 degrees Celsius.
Abt finished fourth on the track behind BMW's Alexander Sims, who was then penalised for causing a collision with Venturi's Edoardo Mortara.
France's reigning champion Jean-Eric Vergne retired after a collision at the start.
The series, in its fifth season, has now had three different winners in three races.
(Reporting by Alan Baldwin in London, editing by Peter Rutherford)
