Norwich boss Paul Lambert told BBC Radio Norfolk Rose's experience was a key factor in his signing.
"Michael's played a lot of games and knows what it's like to get a promotion," he said.
"He's had a great up-bringing at Manchester United, you don't get any bigger, and he's done it the hard way.
"He's been to clubs on loan, he's been up at Stockport with the situation up there and he's been one of their best players," he added.
Rose will stay with the Canaries until the end of the season and will immediately deputise for Drury, who Lambert says will be sorely missed.
"My view is that he's been one of the best players for me, it's a blow to lose him, but it's unfair to compare them. What he [Rose] will do is give you everything he's got, I know that for a fact," he said.
Fellow left-back Rhoys Wiggins has joined Bournemouth on a month's loan, with Lambert confirming that he hadn't played enough first-team football this season.
"Rhoys has had a stop-start season due to injuries and not many reserve games, it would be a massive ask for somebody as young as that to go into your back-four the way things are going," he said.