Use a sprite as your cursor, control it with the gamepad object. Check to see if the sprite is over the draggable object when the player hits the button. If it is, attach the object to your controlled sprite. And release it (set its x and y) to the last position the controlled sprite was at then the button was hit.
The technique @BluePhaze said should work fine in both CC and C2, what he's referring to as "IE" is Internet Explorer, for when you export a game from Construct 2. At the moment Construct 2 gamepad should only work in Google Chrome web browser I believe.
Construct Classic should work fine for 360 controllers as long as the game itself runs (meaning the computer you run it on has Windows and DirectX9 (free) usually)