Milan go into the game in 11th place in Serie A, a massive 21 points behind league leaders Napoli, meaning the Italian Cup and the Europa League, where they have made the last 32, are their only avenues to silverware this season.
“This (Wednesday's game) is like a World Cup final for us,” football-italia.net quoted Gattuso as telling Milan TV.
“Nobody expected this many difficulties, but this derby can change our season on both a psychological level and in terms of results.”
Inter are third in Serie A, five points off the lead after suffering back-to-back defeats against Udinese and Sassuolo.
On Tuesday, Lazio made the semi-finals with a 1-0 win over Fiorentina thanks to a goal after just five minutes by Bosnian midfielder Senad Lulic.
Lazio will face either Milan or Inter in a two-legged semi-final in January.
The remaining quarter-finals between Napoli and Atalanta and Juventus at home to city rivals Torino will be played at the start of next month.