CJam, 11 bytes
OK, let's go with the obvious trivial solution. At least it's shorter than the Python and Ruby solutions that use the same trick.
{N\"_~"}_~
This code begins with a newline, which is ignored by the CJam interpreter. All the actual code is on the last line, which is not changed by the unfurling process. The extra spaces and newlines introduced by unfurling the empty first line are also ignored by the interpreter.
Otherwise this is just a trivial tweak of a (or perhaps the) standard CJam quine. {N\"_~"} defines a code block, _~ makes a copy of it and executes it. Inside the code block, N\ pushes a newline before the copy of the block on the stack and "_~" pushes the string _~ after it. When the program ends, the CJam interpreter automatically prints everything on the stack, i.e. the newline, the code block and the _~.
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