if you are in 2nd place when the other player crosses the finish lane of his final lap, you can do this.  i haven't tested this yet, but i'm almost sure this is how to do it.  before the other player finishes, drive into a red shell, green shell, peel, or black ball (this might work if the other player uses a lightning and may be difficult for certain items).  if you have a shroom in your inventory, use it as soon as you start to spin out.  you will be able to drive until you finish your final lap.

http://membres.lycos.fr/duffjr/matchglitch.avi