Kamara hits first goal for Fulham in Cup win
The Senegal striker had seen a header earlier come back off the bar following good work from David Healy.
Fulham boss Lawrie Sanchez: "It was good for Kamara to get off the mark. He has been close to doing so and that will be an important moment for him.
"I have to be pleased with the win as Shrewsbury are a good team and well organised.
"It is credit to their manager Gary Peters, who I know from our days together at Reading.
"I thought we controlled the game but we did not take our chances and that was a bit of a worry. We have looked a bit vulnerable in the last five minutes this season."