Cacau opened the scoring in the 13th minute at the far post when the turned in a cross by Christian Molinaro, and it was 2-0 on the half-hour when he hit a sublime drive beyond Faryd Mondragón.
He wasted no time completing his treble, increasing the lead to 3-0 in the 38th minute after tucking the ball home from a goalmouth scramble.
Christian Schorch pulled a goal back for Köln just before the break with a header past Jens Lehmann, but Stuttgart completed the rout in the second half.
Pavel Pogrebnyak got on the score sheet in the 70th minute when the Russian nodded home Roberto Hilbert's cross, restoring Stuttgart's three-goal advantage at 4-1.
Cacau then completed his four-goal haul four minutes later when he tucked home after Pogrebnyak's service.