That is because left and right side of the body is not totally symmetrical. If one arm gets tired much sooner than the other it is best to use a barbell. With a barbell, the stronger arm will compensate the weaker and in time both sides will become more symmetrical in strength and then you can use dumbells.
It worked for me.
It is best to have someone next to you so he/she can monitor your form. In moments when the weights are heavy, we all tent to twist our bodies just to keep those weights moves. Or it is a good idea someone to make a video clip while you are doing the exercise so you can then see if you are doing anything wrong.
It was more of a pain in the center of my palm rather than muscle soreness but a barbell might help to give that arm more stability. I'll give it a week to recover and see what happens next time.