Serie A wrap: Sampdoria shock AC Milan
In the shock of the opening weekend of Italy's top tier of football, newly-promoted Sampdoria upset last year's second-placed club despite Milan having the majority of possession at home on Sunday.
Milan had 60 percent of possession and took 23 shots for the match, with six forcing a save from Sampdoria's goalkeeper Sergio Romero.
But it was the visitors from Genoa who took the points when Costa leaped highest in the 58th minute to head Fernando Tissone's corner inside the far post.
Milan's city rivals Inter had a much more positive start to the season on Sunday, winning 3-0 over Pescara.
Inter took the lead away from home when Wesley Sneijder tapped in Diego Milito's cross in the 17th minute.
Just two minutes later, Inter made a mockery of the defence of last season's Serie B champions when Sneijder released new signing Antonio Cassano down the left who crossed to Milito for a simple finish.
Inter wrapped up their victory in the 81st minute with another goal from around the six-yard box, Milito setting up substitute Philippe Coutinho to lift the ball into the roof of the net.
That result put Inter to the top of the Serie A table where they share the lead with Napoli who defeated Palermo 3-0.
Napoli went ahead in first-half injury time thanks to a goal from Marek Hamsik.
The Slovak midfielder set up Christian Maggio for Napoli's second goal in the 79th minute, while Edison Cavani completed the impressive victory away from home with a header two minutes from time.
Roma had to rely on a 91st-minute goal from Nicolas Lopez to salvage a 2-2 draw at home against Catania, while Lazio defeated Atalanta 1-0 and Chievo won 2-0 over Bologna.
In other Serie A results, Genoa defeated Cagliari 2-0 and Siena played out a goalless draw with Torino.