Matias Galarza's goal, scored a tick over one minute into the clash between two of Australia's Group D rivals, proved the difference in Santa Clara.
The result means victory in the match between the Socceroos and Paraguay, kicking off at 1200 AEST on Friday, will be rewarded with second spot in the group.
A draw between Australia and Paraguay would leave both sides level on four points, but Tony Popovic's side would finish second because of their superior goal difference.
The group's third-placed team will still have a chance to reach the knockout phase, while Turkey are guaranteed to finish fourth and head home after their dead rubber against the table-topping United States.
A 10-man Paraguay held on desperately during a frenetic end to the game, which was marred by Almiron being given his marching orders during first-half stoppage time.
The absence of the star playmaker, who plays for Major League Soccer club Atlanta United, will boost Australia's hopes of straightforward progression to the round of 32.Â
Almiron and Mert Muldur exchanged words following a foul in the middle of the field.
Almiron covered his mouth while saying something to Muldur, who immediately appealed to referee Ivan Barton for punishment.
Barton went to video review and quickly ruled Almiron would be given a red card and sent off under a new rule put in place for this year's World Cup.
FIFA president Gianni Infantino pushed for the new rule after Benfica's Gianluca Prestianni tried to hide verbal insults toward Real Madrid's Vinicius Junior in a Champions League game.
Galarza gave Paraguay a second-minute lead as he collected a pass from Julio Enciso and fired a low effort into the bottom corner from 20 yards.
With both sides having lost their opening game, there was plenty at stake at the San Francisco Bay Area Stadium and the tension soon led to a fiery encounter.
Muldur's header from a free-kick hit the crossbar and post before bouncing away as Turkey came close to a 33rd-minute equaliser.
Given their numerical advantage it was no surprise that Turkey dominated the second half, with Orlando Gill saving from Abdulkerim Bardakci.
Enciso fired a good chance wide on the counter attack while Can Uzun missed from six yards out at the other end.
Needing a goal to maintain any chance of progressing, Turkey became increasingly desperate as Gill denied Uzun and Deniz Gul put the rebound off target.
Merih Demiral headed a stoppage-time effort wide and, soon after, Turkey's World Cup campaign was effectively over.
