Andreescu beat Serbia’s Petra Marti 6-4, 6-4 in the women’s tournament.
The product of Mississauga, Ont., faces top-seeded Iga Swiatek in the next round.
Swiatek is the defending champion in Rome and is attempting to win her fifth straight tournament.
Montreal’s Auger-Aliassime dispatched American Marcos Giron 6-3, 6-2 in the men’s third round.
He will face world No. 1 Novak Djokovic in the quarterfinals.
Djokovic beat Switzerland’s Stan Wawrinka 6-2, 6-2 on Thursday.
Denis Shapovalov of Richmond Hill, Ont., was scheduled to face third seed Rafael Nadal of Spain later Thursday.