Yeap, exactly that; this will solve the vertical scroll problem.
However, you will still encounter a "cut off" or repetition of the graphic on the horizontal.
Those with high resolution screens 1920px+ wide, will either see a repetition in your graphic, or a sudden cut off to a colour (your main BG colour).
You could always make the graphic 1920px wide, but that will create a rather large file size.
You could blend the edges of the graphic into a colour (your main BG colour), so that there is no cut off.
Or you could modify the graphic so that it repeats well on the horizontal axis.
Varies ways of the doing it, looking good though
Example:
http://www.deckhandmarine.com/