Considering that early Homo Sapiens arrived at Australia some ten thousand years earlier than he entered Europe, there must have been something that kept these people from doing so.
Most probably not the Neanderthals, but the weather...
They came out of Africa as nice weather guys, and it took them some time to adopt to Europe's Ice Age climate. Probably they learned from the Neanderthals...
And then, when they proceded into Europe, they brought with them some nice African germs for which the Neanderthals had no anti-bodies. End of the Neanderthal story. No mass murder scenerio required. Just the poor big noses sneazing themselves to death.
The Neanderthals, on the other, seem to have suffered from the extreme cold as well, so their population base was already rather small when the newcomers arrived.
To change this: Neanderthals keep away from contact zone with Sapiens after some initial infections occur. (They of course do not identify Sapiens as perpetrator but think area is bad - this also leads to some immunisation in the Neanderthal population) Thus Sapiens doesn't learn to adopt to cold weather. Neanderthals in Europe recover and hold out until end of Ice Age. When Sapiens tries again, there's already a well developed Neanderthal hunter-gatherer society established, all hunting grounds taken, Sapiens turn east.