In the other Group D game, played earlier on Monday, Panama beat Bolivia 2-1.
Argentina posed a constant threat with fast sweeps forward and Di Maria capitalised on one such attack after 50 minutes, running on to a pass from Banega to slot home from eight yards.
"We've been doing things properly for years now and maybe this year it will come right," said Di Maria, who broke down in tears after dedicating his goal to his grandmother.
Argentina doubled their lead eight minutes later in similar fashion, this time Di Maria feeding Banega in the same position on the left and his deflected drive beat Chile goalkeeper Claudio Bravo at his near post.
"If you make mistakes in these games you pay a high price and that is what happened today," Bravo said in a televised interview after the game in Santa Clara.
Second-half substitute Erik Lamela twice came close to adding to Argentina's lead but it was Chile who got the third goal of the game thanks to a mistake from Sergio Romero.
The Argentine goalkeeper, who had an otherwise excellent game, misjudged a free kick in the box and Fuenzalida nodded the ball into an empty net.
The result gives Argentina, who are seeking their first major title since lifting the Copa America in 1993, a measure of revenge for their defeat to Chile in last year's Copa America final.
Also in Group D, veteran striker Blas Perez scored twice as Panama beat Bolivia.
Perez scored his first after 10 minutes when he slid in to push home a cross from 10 yards out before Juan Carlos Arce tied the score eight minutes into the second half.
Perez grabbed the winner three minutes from time.
Panama next play Argentina on Friday while Chile face Bolivia.
(Writing by Andrew Downie; Editing by Peter Rutherford)
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