The victory, secured by goals from Angel di Maria and Gabriel Mercado after Felipe Gutierrez put the home side ahead, was Argentina's second of the campaign after winning just one of their previous four games.
The loss of Arturo Vidal was felt in the middle of the park for Chile, as the Bayern Munich midfielder's absence left the home side short of power.
Starved of possession, Alexis Sanchez and his fellow forwards could not trouble an undermanned Argentina defence, which was missing Javier Mascherano.
Di Maria produced a moment of pure magic when he levelled the scores after 20 minutes, providing an inch-perfect finish from the left side of the box to leave Chile goalkeeper Claudio Bravo helpless.
His goal was well-timed with his side having to recover from going a goal down after Gutierrez ensured Chile's good start was not in vein when he sent home a thumping header from a Fabian Orellana corner after just 12 minutes.
It was Mercado's effort that proved the difference in the end, as he got on the end of a lovely Lionel Messi flick to volley home the eventual winner.
The home side had enjoyed the better start, despite losing Matias Fernandez to injury after just seven minutes, as their visitors struggled for inspiration in the final third.
Gutierrez made good on that start with his header after 12 minutes as Argentina were left reeling.
Unsurprisingly, it was Messi who looked most likely early on, picking up the ball in his own half after nine minutes and driving towards the backpedalling Chile defence before picking out Di Maria, who skied his shot.
But Di Maria would not miss twice and when he was found unmarked in the box by Ever Banega, the Paris Saint-Germain winger curled a brilliant effort home to level after 20 minutes.
Chile lost another player to injury when Marcelo Diaz was taken off, while the equaliser sparked greater confidence from Argentina and they took the lead minutes later through River Plate defender Mercado, who athletically tucked away a Messi flick for his first international goal.
Sanchez went closest to equalising after the restart as his lobbed effort went just wide, before Sergio Aguero sent a fierce curling effort off target in the 64th minute.
Chile rallied hard in the closing minutes, with Sanchez going close from a late free-kick, but could not find the back of the net for the all-important equaliser.