A spokesperson for the Spanish capital's emergency services said the 25 people had been treated for "cuts and bruises, all minor".
The altercations happened as several buses carrying Real Madrid fans arrived at Atletico's ground, with home supporters greeting them by throwing bottles and other objects.
Police were forced to form a barrier between the rival fans, keeping them apart to prevent further trouble.
None of those injured required hospital treatment and police reported no arrests.
Reigning European champions Real came into the game leading 3-0 after last week's first leg.