nozturk,
The two are roughly the same, but not exactly the same in meaning.
1) I've got this situation in hand. = I've got this situation under my control.
2)I've got this handled. = I am controlling the situation or having someone else control the situation.
2) is the "get" passive. "To get" also can function in your sentence as a causative verb.
I've got this handled. = The situation has been handled. (by me or by someone as my agent)
Mighty Mouse ( a famous American cartoon character) used to sing the following song:
Here I come to save the day!
That means that Mighty Mouse is on the way!
On the sea or on the land,
He's got the situation well in hand!