This is a photoshop thing, so it might not work the same in SAI, but as a continuation of Mimeme's advice, if you do that thing with the wand tool, you might get a very slightly smaller area selected than the shape of the figure. This is due to there invariably being some near-invisible grey pixels the wand tool won't select. It'll leave an ugly pixely border of uncolored pixels by the edge of the figure. If this happens, you can (probably) expand the selection so that the selected area go "under" the lines. Most likely, this option would be in the "Selection (S)" menu, and would be called something along the lines of "Expand".
So, wand tool click inside the lines, selection -> expand by a few pixels, and then color.