Nacional, coached by Gustavo Diaz, progressed 4-2 on aggregate.
Alexis Rolin put Nacional ahead in the sixth minute with a header, before Gonzalo Bueno doubled the lead from close range just before the interval after good work by Alexander Medina.
In the 83rd minute, Iquique midfielder Ruben Taucare was sent-off after a foul and Alvaro Recoba converted that free-kick to extend the margin.
The win was wrapped up in injury-time when Matias Vecino netted, seconds after midfielder Santiago Romero was dismissed to even the numbers up.
"We maybe felt tired in the second half because we tried to impose our speed in the first," Recoba told Fox Sports.
"Iquique were tough rivals and we may have a better squad on paper, but we had to show it on the field. I think we won deservedly."
Ecuador's Liga de Loja or Monagas of Venezuela will be Nacional's next opponent.
In Colombia, host Millonarios booked a berth in the next round thanks to a 3-0 victory over Inti Gas Deportes of Peru.
Having drawn 0-0 in the first match, Wilberto Cosme put Millonarios ahead in the 36th minute by heading a Lewis Ochoa cross into the net.
Harrison Otalvaro set up Wason Renteria to make it 2-0 in the 72nd minute, then scored himself in the last minute of the game, as the 2007 semi-finalist set up a tie against Paraguay's Guarani for a place in the round of 16.
In Uruguay, Cerro Largo played out a goalless draw with Bolivia's Aurora, which progressed thanks to a 2-1 first leg win.