Problem with powering 12V RGB strip and Arduino Uno from the same source.

What you say you did sounds reasonable, however it is often the case that you missed something. So a schematic and photograph would be a good start.

It could be your power supply, did you have a large capacitor across it?
Did you use solderless bread board? It is not good at handling large currents. How long is your LED strip and what current is it taking?