Ive done it a bunch of times and a good Iron 25 watts, and a de-solder pen. I usually put a tiny bit of flux on the solder points before i start just to get the stuff flowing. You have to be quick. Heat , and then suck really quick. Once all the solder is sucked off the pins, clean your iron tip and just go around and touch every pin as you pull on the analog stick from the other side of the board. Then Like Matt said besure to clean the solder pads with a wick , and re-tin your new stick. do not put to much solder on the pins for the new stick , this will make it more difficult to put back in the pin holes. The just go around and solder up the stick to the board.
NOTE: add to video tute list.