Juventus, likely to be Inter's main challengers again this term, fared better with Vincenzo Iaquinta powerfully heading home debutant Diego's free kick in a 1-0 win over Chievo in Turin.
Brazilian playmaker Diego, a new arrival from Werder Bremen, was lively throughout with Fabio Cannavaro making a composed return to the Juve lineup and avoiding any boos from those still angry about his departure to Real Madrid in 2006.
Juve coach Ciro Ferrara, who replaced Claudio Ranieri towards the end of last season and secured second spot, endured some nervous moments late on but ended up much happier than Inter's Jose Mourinho.
Cameroon forward Eto'o, signed from Barcelona in a swap deal with Zlatan Ibrahimovic, slotted in for Inter after Diego Milito was fouled by Leonardo Bonucci in the 56th minute.
The hosts had much the better of the play but their defence, including new signing Lucio, fell apart to allow Vitali Kutuzov to prod home on 74 minutes for the promoted side.
BLASTED OVER
The forward from Belarus almost won it for the visitors but Julio Cesar pulled off a fine save before Bari's Emanuel Rivas also blasted over.
Next Saturday Inter clash with AC Milan, who won 2-1 at Siena in their first game of the season on Saturday.
Inter midfielder Thiago Motta, who signed from Genoa alongside Milito in the close season, was not overly worried.
"We know it was the first game against a team that wanted to play football and they gave us problems. We had chances. We made mistakes," the Brazilian told Sky television.
"Next week is very important for us and we'll get to work. But it is only the start of the championship and there is no anxiety."
Argentina's Milito twice went close in the first half and linked up well with Eto'o, who spurned a glorious chance after the spotkick.
Genoa, fifth ahead of AS Roma last term, beat their rivals 3-2 in a thrilling game and promoted Parma drew 2-2 at Udinese.
Fiorentina, who finished fourth in May, warmed up for Wednesday's Champions League playoff second leg against Sporting with a 1-1 draw at Bologna on Saturday.
