As stated above, you'll go through passport control in Copenhagen and customs (if at all) in Rome. Your flight into Copenhagen will arrive in one of the terminals for non-Schengen flights, and your flight to Rome will depart from one of the terminals for Schengen flights. To get from one to the other, you'll pass a desk where you'll get your passport stamped.
Passport control for entering or leaving the Schengen zone is not a complicated process. If you're thinking of the long lines, paperwork, and questioning you have to endure to enter the US, UK, or Canada, it's nothing like that. You might have to wait in line for maybe 5 minutes. You probably won't be asked any questions.
CPH is not a large airport, they pride themselves on running things efficiently, and they're used to facilitating tight connections. If your layover is at least 40 minutes (the minimum for connecting from non-Schengen to Schengen, as listed here), you'll be fine.