Well, I am not sure about containment, but what if the Mongols do not reach Caffa and the plague isn't introduced.
Then the plague would have been carried by trade, from Levant and Egypt.
Sorry, I had my dates wrong. I was wondering what would happen if they had been hit at the same time, I thought the Muslim world wasn't hit until after the European outbreak was over. so instead, what might happen if the Siege of Caffa doesn't occur, the infection of the Muslim world doesn't spread to Europe, but the news arrives in Europe. Might they launch another crusade believing God has weakened their foes and infect Europe anyway?
Actually, European have news about plague in Orient. There's mention of "Tartaria devastated, Mesopotamia depopulated, etc."
Now, in the short time where Middle-East was touched and Europe not, yes technically they could use it.
But ask yourself this question : WHO would be idiot and sucidial enough to go in a land where an epidemy kill mass of people? It's not because it's the Middle-Ages that they didn't understand this : when a country is ravaged by an epidemy, better being far from it.
Merchants could still go, because european need their imported goods, but they avoided contacts with population.