Waves on strings combine linearly. This means that you can split up a string's motion into two (or more) superimposed waves. The two superimposed waves behave independently, as if the other one was not there. So if you have a standing wave set up on a string, and then you also introduce a travelling pulse, you get something like the following. (The arrows represent the direction of movement, and the node is marked with a blue dot.)
Now to answer your question. I wish I had a way to make the picture animated, but I think you can see it from still pictures. I'm going to draw what happens after a short time, when the pulse reaches the node. The standing wave has also moved, and is now swinging back in the other direction.
As you can see, the standing wave component still passes through zero at the node, as it always must, but the combined wave (pulse + standing wave) does not. Because the pulse and the standing wave do not interact, the pulse just passes straight through the node as if it wasn't there, and the standing wave just keeps waving as if the pulse wasn't there.
Note that not interacting isn't the same as not interfering. Interference happens when two waves get added together and sum to zero, but neither of the two waves is affected by being added in this way, so even when waves interfere, they don't interact.