Paris Saint-Germain's star-studded side lost for the first time this season as Strasbourg held on for a famous 2-1 victory following Stephane Bahoken's second-half strike.
After 13 wins and two draws to start the league season for Unai Emery's pace-setters, Saturday's result came out of the blue.
Da Costa stunned the visitors with a 13th-minute opener, getting in behind their defence to nod home a free-kick swung across from the left by Dimitri Lienard.
Kylian Mbappe pounced for an equaliser in the 42nd minute, the teenage striker meeting a driven cross from the left by Adrien Rabiot at the near post and giving Strasbourg goalkeeper Bingourou Kamara no chance.
But Strasbourg restored their lead in the 65th minute when Bahoken burst through the middle to beat Alphonse Areola with a thumping strike high into the net.
Home shot-stopper Kamara had to be replaced due to injury with 20 minutes remaining, as Alexandre Oukidja came on in his place, and moments later the league's leading scorer Edinson Cavani stepped off the bench as PSG chased an equaliser.
Pressure was intensely applied to the Strasbourg goal, with nine minutes of stoppage time giving them even greater hope of a leveller.
But Neymar was booked as frustration in the PSG ranks grew, with Strasbourg clinching a stunning win.