Can a goalkeeper score a goal with his hands?

Of course it is a goal. The goalkeeper playing the ball with his/her hands is no different than if he/she played it with his/her feet.

Why wouldn't it be a goal?
If the ball is in play, and if he/she has the ball legally (not from a deliberate kick from a teammate or from a throw-in by a teammate) the goal is valid.

Heck of a throw by the goalkeeper.