You can warmonger just like the U.S. They have not taken any cities, just pillaged and destroyed the military. Civ loves that approach.
Plus, in real life the US is good at getting other countries to join them in their wars - which helps to reduce diplomatic backlash. The exact same thing applies to Civ as well.
Seriously, this does help - particularly after the most recent patch as your warmonger penalty will be halved with Civs who are at war with the same enemy as you. Try bribing other Civs to attack the Civ that you want to attack if you can afford to do so.
Also, denouncing a Civ before you attack them will reduce the initial warmonger penalty somewhat - sneak attacks without warning are frowned upon. Also, it's
really not a good idea to declare on somebody who you have a Declaration of Friendship with.
Some other factors that help lessen diplomatic penalties are having a shared Religion or Ideology or having a Declaration of Friendship. And if you liberate a City State or another Civ's city that had previously been taken by your enemy you can wipe out a LOT of the warmonger penalty.
Lastly, the AI has four levels of warmonger hate and different Civs will have different tolerance levels to warmongering. The Huns will tend to 'overlook modest warmongering' for instance, so you will be able to get away with a lot where they are concerned but India will hate even modest warmongering and will grow to dislike you very quickly when you go to war.
Just some tips that I hope will help you to 'stack the deck in your favour' when pursuing a military campaign.