The victory completed an incredible rise back to the top for a club which was relegated to the fifth tier in 2011 after financial difficulties.
It was their fourth League Cup triumph, to add to three French Cups and the 1979 Ligue 1 title.
Neither side managed to create much in normal time, although Ludovic Ajorque did force Guingamp goalkeeper Marc-Aurele Caillard into a fine 89th-minute save.
At the end of the added half-hour it was Guingamp who almost snatched the trophy, but Alexandre Mendy mishit an effort straight at Strasbourg ‘keeper Bingourou Kamara.
Mendy blazed over with his team’s first penalty to gift Strasbourg the advantage, and Kamara saved Rony Rodelin’s weak attempt before Lionel Carole secured a 4-1 shootout victory.
Guingamp, who ended Paris Saint-Germain’s bid for a sixth consecutive League Cup in the quarter-finals, will now turn their attentions to avoiding relegation from Ligue 1.
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