Argentina opted to rest Messi after his COVID-19 infection, but his Paris Saint-Germain team mate Di Maria stepped up to give the visitors the lead with a trademark curling shot from outside the box after 10 minutes.
Argentina then took advantage of an injury to Chile goalkeeper Claudio Bravo to seal the win 11 minutes before half-time.
Bravo was preparing to be substituted when Rodrigo de Paul fired in a fierce drive from 30 metres, and with the goalkeeper only able to parry the shot Lautaro Martinez followed up to slot home the rebound.
"We knew it was going to be difficult at high altitude and after a long flight," said Di Maria. "The important thing is that we come away with a win.
"Everything is always much easier when Leo is here. The important thing is that we keep working like we've been doing and bringing joy to Argentines."
The result, which was also achieved without the presence of talismanic coach Lionel Scaloni, who was isolating at home due to COVID-19, marks Argentina's 28th game without defeat.
Brereton's goal was the first against Argentina in seven qualifiers.
"Today we were without the best in the world and our coach as well but we got the victory and this is also for them," said Di Maria.
The result was a huge blow to Chile's hopes of qualifying for Qatar. They are seventh, on 16 points, in the 10-team South American qualifying group with only three matches to play.
The top four qualify automatically and the fifth-placed team goes into a playoff with a side from the Asian confederation.