Step 1: Do you really want to do this? Do you think you can do this?
No offense, but if you are having to search this topic, I worry about your abilities to follow through. I mean, you obviously have hesitated and the opportunity is probably gone. You have lost the upper hand. Still, read the tips and retain them for the next "episode." However, chances are you have already lost this one.
Step 2: Look in the mirror.
Take a second look at that poor sap in the mirror. Are there bloodshot eyes? Nose clean? Make-up out of place? If so, do a quick fix-up. Three-year-olds are vicious and they will go for any perceived weakness to change the subject. "You have cracks in your eyes." "What is that black on your face?" All familiar derailing techniques. Do not give him/her the opening.
Step 3: Deep breath.
This is going to be worse before it gets better. I'm here for you. Take solace in that. Not too much solace, because I can't actually "win" this for you, but maybe level "yellow" solace in a sliding rainbow scale.
Step 4: Verify that you are, in fact, disagreeing.
I cannot tell you how many times I have argued with my 3-year-old, only to find out that we were actually in agreement. Embarrassing for all involved (though admittedly she doesn't really show any signs that it bothers her). For example: She says "turn up" the radio, but I want it turned down. We bicker, then I turn it down further and she says, "See, that better -- I say turn it up." At this point I could turn it into a disagreement about how she needs volume control instructions, but it's better for both of us to feel like it's a win.
Step 5: Be the bigger person.
Both figuratively and literally. I touched on this above when I said I made like Elsa and "Let It Go" when it came to the volume. Additionally, don't slouch. Tower over the child. Assert dominance. Wait a second. That's called being a pack leader of a wolf pack. Maybe literally is not as important.
Step 6: Prepare for tears.
Have Kleenex on hand. We hope this is for the 3-year-old when he/she finally feels the soul-crushing feeling of being wrong and the realization that mom and dad are totally amazing for looking out for his/her best interests all along. The other thing that could occur (not to scare you, but you should be prepared, just in case)... You could have a small weepy breakdown from answering "why" for the 15-bazillionth time. Chances are small, but I need to be honest with my readers.
Step 7: Powerpoint presentation.
Three-year-olds are visual. You can't just talky, talky at them. They want to see pictures. It can be vacation pictures or graphs. Doesn't matter really, just have a presentation available.
Step 8: Make up a rap song.
Have the child beatbox for backup. Bonus points if you comment below with said rap song. Keep the swearing to a minimum... or not -- your kid, your choice.
Step 9: Enlist backup.
I find favorite stuffed animals are great for backing me up in arguments. "I'm Princess Sparkle Rainbow Bear and I support the message brought to you by your mommy." Practice throwing your voice before trying it in case it's above your current station. Ventriloquism is a skill you should have, though, so even if you aren't good yet -- practice up. It is only a matter of time before the skill will come in handy outside of parenting. Dark alleys, doctors' offices, and funerals, to name a few situations.
Step 10: Make faces and move on.
Redirect. Tell them I'm the mom and batten down the hatches. Did you really think that you were going to win an argument with a 3-year-old? In my limited experience, they never admit defeat. Keep that innocence, though, and stay sweet.