There was disappointment, however, for fourth seed Elena Dementieva, who became the highest-ranked casualty when she suffered a 7-5, 6-4 defeat against Gisela Dulko of Argentina.
Dementieva, 23, withdrew from last week's Berlin Open with a hip injury but was considered one of the favourites in Rome after reaching the finals at Roland Garros and Flushing Meadow in 2004.
From the start, the Russian found herself being dragged around the court by Dulko's precise and consistent groundstrokes.
Having lost the first set, Dementieva dropped her opening service game of the second to slip 3-0 down.
Dementieva found her timing to break back but a pair of errors as she served to stay in the match gave Dulko two match points, which the Argentine converted when her opponent dumped a backhand into the net.
The world number five followed compatriots Svetlana Kuznetsova and Elena Likhovtseva out of the tournament.
Second seed Mauresmo, who received a bye in the opening round, was barely tested by Stosur.
"I felt it was close at the beginning, then I put my level up a little higher. 6-2, 6-0 is a great way to start the tournament," said the Frenchwoman, who has appeared in four of the past five Rome finals.
"I'll try to get that trophy again but it's going to be tough. Obviously I'll just take it match by match."
She was joined in the last 16 by 15th seed Ai Sugiyama, who beat Venezuela's Maria Vento-Kabchi 6-3, 6-0.
Later, Australian Open champion Serena Williams takes on Italy's Francesca Schiavone and Nadia Petrova faces another home player, Mara Santangelo.
