What someone believes in typically based upon facts or stats.
So yes you can win an argument with only opinion, because opinion is a folder that contains facts and stats. Whether or not somebody agrees with you or not, is irrelevant to the truth.
For instance: Two plus Two equals Four. It is an opinion, based on accepted beliefs, that two and two make four. You can also say it is a fact. However, that does not mean somebody can't argue you and say "No, two and two equals five."
Have you ever once in your life, heard of a case in Law where, during the closing arguments, either the Defense or Prosecuting attorney said "People of the jury, your honor, the other lawyers evidence was so damn good that I agree with them"
Nope. Both attorney's will use the closing argument to try and continue their point, like a bunch of stubborn little kids.
An argument for why two plus two equals five. Is that, in this person's understanding, two actually means two and a half of one. Both people have their own opinion, yet both see it as a fact in their mind.
The only way to convince the person that two and two is four. Is to either write it numerically "2" in order to prevent the defining of "two" to be based on an individuals accepted understanding. If that does not work, one must use physical pieces, and yet, one could still hold an opinion that you are wrong and they are right.