Most American Airlines allow traveling with Carry-on dogs. Usually one dog is allowed per ticket and there is a maximum number of dogs that can be on-board a flight and so you need to get your dog registered well in time. This LINK may provide you some further assistance.

 I used to travel for work every week and most airlines I flew allowed small pets as carryon. I do not know the details on this and do be careful as some airlines use a commuter airline at times. These smaller planes can make it very difficult to get a standard size carryon case under the seat (no rollerboard bags for sure).