I have done this
many times. Well done!
How I did it: I did it by being nice! My young children and I were traveling across the US to see my family. Our first flight was late and we missed our connection. Everyone was instructed to see the attendant at the gate to get our new flight information and after a solid 20 - 30 minutes of watching the other passengers scream into this woman's face it was my turn. I'm a pretty laid back person and didn't mind waiting but the one thing I really didn't want was to be split up into different parts of the plane from my kids. I told her that I realized it might be tricky to seat us together but I would really very much appreciate being seated with my children, even if we had to wait. We weren't in a terrible rush and of course no one likes waiting but we just wanted to sit together because flying was a little scary for us. I think she was so shocked that I wasn't screaming that she very sweetly told me she would find something for us. And in minutes she had us upgraded to first class on the next flight out. Nobody that yelled in her face got onto the flight with us.
Lessons & tips: Be nice!