Bari, currently struggling in the relegation zone, took all three points at the San Siro when Masinga smashed in a rebound after the Inter keeper Gianluca Pagliuca had parried his header.
The Milan side paid dearly for a second league defeat of the season, their lead at the top narrowing to just one point over Juventus, who beat Bologna 3-1.
Juve's Filippo Inzaghi scored twice within the opening 20 minutes and Alessandro Del Piero thumped home a 25-yard free-kick on the hour. Igor Kolyvanov scored a consolation goal in injury time for Bologna, who were without Roberto Baggio, following an argument with Renzo Ulivieri before the match.
The Bologna coach had asked the striker to go on the substitutes' bench. Baggio refused, then left the ground and went home. The Bologna president, Giuseppe Gazzoni Frascara, said afterwards that the club fully supported the coach and they may consider selling the player. "It depends on who wants him. Ulivieri made a technical choice and Baggio refused that choice," he said.
Fiorentina's 2,000th match in Serie A did not give them any cause for celebration, as it ended with a 3-1 home defeat to Lazio after the former Arsenal midfielder Stefan Schwarz was sent off in the 48th minute.
In France, the defending champions, Monaco, and league leaders Metz both avoided first-round embarrassment against opposition from lower divisions in the French Cup on Saturday.
Bosnian defender Muhammad Kojnic scored the winner seven minutes into extra time after Monaco had been held 1-1 for 90 minutes by Fourth Division Dijon.
Metz struggled even harder to beat Second Division Le Mans, the game going to a penalty shoot-out in which the Metz goalkeeper, Bruno Letizi, twice saved timid spot-kicks.
Strasbourg, who beat Rangers, Liverpool and Inter in the Uefa Cup this season, were defeated 2-1 by Epinal after having Jean-Luc Dogon dismissed in the first half.
Reports in Spain say that Louis van Gaal, the Barcelona manager, has turned down an offer to coach the Dutch national side.