Morgan tore a chest muscle in the Dragons' Amlin Challenge Cup defeat to Exeter Chiefs on Sunday and is expected to be out for four months.
The Dragons welcome Exeter to south Wales this weekend in the return fixture and head coach Darren Edwards knows it's a must win game for his side.
"You have got to win your home games in Europe if you are going to have any chance," he said.
"They played a pretty heavy kicking game down there which was limited but very effective.
"So its up to us to put the right strategy in place this weekend.
"Let's play some positive rugby on Sunday and end up getting the better of them."