When your ball is is deep rough around the green, the secret to escaping is to have your club make contact with as little grass as possible. That means you need to set up for a steep downswing.
With sand wedge in hand, set up with the ball further back in your stance than normal. Open the clubface and aim to the right of the target. Shift about 60% of your weight on your leading leg. Flex your knees to keep the weight low. Holding onto the grip more tightly than usual, take the club up steeply, cocking the wrists early.
All of this will promote the steep downswing you need to keep the grass from grabbing the face and hosel.
It’s likely that you will need only about a 3/4 swing to get the job done. Hit down sharply and hard. Keep a tight grip to prevent the grass from grabbing the hosel and closing the club face.
The steep, sharply hit downswing will make the ball pop out with some spin, fly high and land softly on the green.
The thing you most want to avoid is bringing the club in low or flat. When it hits the grass, this will cause you to top the ball and send it flying across the green to the deep rough on the other side.