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Helen Richardson-Walsh struck twice in the second quarter to give Great Britain a 2-0 lead at half-time.
Sophie Bray then won a scramble in front of goal to knock home GB's third, before Florencia Habif pulled two back for Argentina.
Group B leaders Great Britain next face Japan on Friday.
Goalkeeper Maddie Hinch pulled off some fine saves as Argentina, who won bronze at London 2012, grew stronger throughout the game.
Richardson-Walsh saw her fourth-quarter penalty flick saved after a foul on Alex Danson, but Great Britain were able to hold on.
"Beating a team like them gives us confidence but I don't think we're firing on all cylinders," said Richardson-Walsh.
"At the start we felt comfortable, which is good, but this tournament is tough. Anything can happen in the quarter-finals but we're building momentum and hopefully we can use it."
Kathleen Bam scored three goals as the United States beat Japan 6-1 to stay second in Group B.
Jodie Kenny struck twice as Australia beat India 6-1, with Anuradha Thokchom scoring a consolation goal a minute from the end.
New Zealand moved up to second in Group A as they beat Spain 2-1 thanks to two goals from Kelsey Smith, while Germany beat South Korea 2-0 and China and the Netherlands played out a goalless draw.
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Great Britain's women maintained their unbeaten record as they qualified for the Olympic quarter-finals with a 3-2 victory over Argentina.