Eto'o broke visiting Parma's stubborn resistance when he curled in a right-foot shot in the 71st minute before Diego Milito wrapped up victory with a late strike to give the Serie A champions seven points from three games.
Inter are two points behind Genoa, Sampdoria and Juventus, who all maintained their 100 percent records this weekend.
Genoa rallied to thump Napoli 4-1 with the help of Hernan Crespo's first league goal since moving from Inter in a match in which both sides had players sent off and Napoli finished with nine men following Salvatore Aronica's late dismissal.
Samp beat Atalanta 1-0 away despite Fernando Tissone's red card courtesy of Daniele Mannini's 64th-minute goal and Juventus won 2-0 at Lazio on Saturday.
Inter, who thrashed AC Milan 4-0 before the international break, went out to attack at the San Siro, with Wesley Sneijder forcing a good early save from Parma keeper Antonio Mirante.
But Parma frustrated the hosts by defending in numbers and Brazil defender Maicon shouted back angrily when sections of the San Siro crowd booed Inter off at the end of the first half.
Coach Jose Mourinho tried to spice things up by bringing on Mario Balotelli in a three-pronged attack alongside Eto'o and Milito.
His entrance briefly added impetus but Parma seemed to be getting the measure of them when Eto'o's flash of brilliance broke the deadlock.
A powerful 89th-minute free kick by John Arne Riise gave Roma their first points of the season after defeats by Juventus and Genoa led to coach Luciano Spalletti quitting.
FINE DRIBBLING
They looked in trouble after Massimo Maccarone made Philippe Mexes dizzy with fine dribbling before rifling a left-foot shot into the top corner to give Siena the lead in the 26th minute.
But Mexes brought the visitors back into it with a 73rd-minute equaliser and their cause was helped when Siena midfielder Paul Codrea was sent off minutes later for a second yellow card.
"The important thing is to have won," former Chelsea and Juve boss Ranieri told Sky television. "We suffered against a really good Siena side who played on the counter-attack and caused us trouble, but it went well in the end.
"Now this group must improve and do what it used to do a few months ago. The players need to find the confidence and tranquillity that only comes from winning."
Fiorentina are level on points with Inter after Alberto Gilardino's soaring header in the 55th minute gave them a 1-0 victory over Cagliari.
Antonio Di Natale scored a hat-trick in Udinese's 4-2 win over Catania and Palermo drew 1-1 at home with Bari in a match which started over 20 minutes late because of heavy rain.
